The Twenty First InternationalConference on DomainDecomposition Methods
INRIA Rennes-Bretagne AtlantiqueCampus de Beaulieu, 35042 Rennes Cedex
June 25–29, 2012
ScheduleAuthor IndexSession Index
Version Date: June 27, 2012
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International Scientific CommitteePetter BjørstadUniversity of Bergen, [email protected]
Susanne BrennerLousiana State University, [email protected]
Martin J. GanderUniversity of Geneva, [email protected]
Roland GlowinskiUniversity of Houston, [email protected]
Laurence HalpernParis 13, [email protected]
Ronald HoppeUniversitat Augsburg, [email protected]
David KeyesColumbia Unviversity, [email protected]
Ralf Kornhuber (Chair)Free University of Berlin, [email protected]
Ulrich LangerUniversity of Linz, [email protected]
Alfio QuarteroniEPFL, [email protected]
Olof WidlundCourant Institute, [email protected]
Jinchao XuPenn State University, [email protected]
Jun ZouChinese University of Hong [email protected]
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CommitteesGeneral Chair
Jocelyne Erhel (Chair)INRIA Rennes Bretagne Atlantique, [email protected]
Taoufik Sassi (Chair)LMNO, University of Caen, [email protected]
Local organization
Leonardo BafficoLMNO, University of Caen, [email protected]
Alain CampbellLMNO, University of Caen, [email protected]
Edouard CanotIRISA, Rennes, [email protected]
Christian DogbeLMNO, University of Caen, [email protected]
Caroline JaphetUniversity of Paris 13 and INRIA Paris Rocquencourt, [email protected]
Geraldine PichotINRIA Rennes Bretagne Atlantique, [email protected]
Local coordination
Edith BlinINRIA Rennes Bretagne Atlantique, [email protected]
Fabienne CuyollaaINRIA Rennes Bretagne Atlantique, [email protected]
Webmaster
Aurelien Le GentilINRIA Rennes Bretagne Atlantique, FranceAurelien.Le [email protected]
Nadir SoualemINRIA Rennes Bretagne Atlantique, [email protected]
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Program Commitee
Aziz BelmiloudiINSA Rennes and IRMAR, [email protected]
Philippe ChartierINRIA Rennes Bretagne Atlantique and IRMAR, [email protected]
Arnaud DebusscheENS Cachan Antenne de Bretagne and IRMAR, [email protected]
Julien DiazINRIA Pau, [email protected]
Thomas DufaudINRIA Rennes Bretagne Atlantique, [email protected]
Erwan FaouINRIA Rennes Bretagne Atlantique and IRMAR, [email protected]
Luc GiraudINRIA Bordeaux - INRIA-CERFACS Joint Laboratory, [email protected]
Laurence HalpernUniversity of Paris 13, [email protected]
Michel KernINRIA Paris Rocquencourt, [email protected]
Stephane LanteriINRIA Sophia-Antipolis Mediterranee, [email protected]
Claude LebrisCERMICS and INRIA Paris Rocquencourt, [email protected]
Patrick Le TallecEcole Polytechnique, [email protected]
Yvon MadayUniversity Pierre et Marie Curie (Paris 6), [email protected]
Francois-Xavier RouxONERA and University Pierre et Marie Curie (Paris 6), [email protected]
Damien Tromeur-DervoutUniversite Claude Bernard Lyon 1, [email protected]
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Sponsors (click on image)
Support (click on image)
CopyrightPictures were kindly provided by Y. Chaux, A. Campbell, C. Japhet
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Session Index
Session Index vii
Schedule 1
P1: Plenary Lecture P1 7Laurence Halpern . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
P2: Plenary Lecture P2 9Geraldine Pichot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
P3: Plenary Lecture P3 11Axel Klawonn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
P4: Plenary Lecture P4 13Marcus Sarkis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
P5: Plenary Lecture P5 15Jin-Fa Lee . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
P6: Plenary Lecture P6 17Clemens Pechstein . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
P7: Plenary Lecture P7 19Hyea Hyun Kim . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
P8: Plenary Lecture P8 21Beatrice Riviere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
P9: Plenary Lecture P9 23Xiao-Chuan Cai . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
P10: Plenary Lecture P10 25Eberhard Bansch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
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P11: Plenary Lecture P11 27Blanca Ayuso de Dios . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
P12: Plenary Lecture P12 29Chen-Song Zhang . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
P13: Plenary Lecture P13 31Ralf Hiptmair . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
P14: Plenary Lecture P14 33Michael Holst . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
M1: Finite Element Packages with Domain Decomposition Solvers 35M1 Schedule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
Frederic Hecht . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37Pierre Jolivet . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38Christophe Prud’homme . . . . . . . . . . . . . . . . . . . . . . 39Abdoulaye Samake . . . . . . . . . . . . . . . . . . . . . . . . . 40
M2: Domain Decomposition for Porous Media Flow and Transport 41M2P1 Schedule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
Oliver Sander . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43Thi Thao Phuong Hoang . . . . . . . . . . . . . . . . . . . . . . 44Frederic Nataf . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45Zhangxin Chen . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
M2P2 Schedule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47Bernd Flemisch . . . . . . . . . . . . . . . . . . . . . . . . . . . 48Paul-Marie Berthe . . . . . . . . . . . . . . . . . . . . . . . . . . 49Jean-Baptiste Apoung Kamga . . . . . . . . . . . . . . . . . . . 50Anthony Michel . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
M3: Finite Elements for First-Order System Formulations of InterfaceProblems 53M3 Schedule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
James Adler . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55Pavel Bochev . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56Fleurianne Bertrand . . . . . . . . . . . . . . . . . . . . . . . . . 57Steffen Munzenmaier . . . . . . . . . . . . . . . . . . . . . . . . 58
M4: On the Origins of Domain Decomposition Methods 59M4 Schedule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
Martin J. Gander . . . . . . . . . . . . . . . . . . . . . . . . . . 61Xuemin Tu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62Francois-Xavier Roux . . . . . . . . . . . . . . . . . . . . . . . . 63Olof B. Widlund . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
M5: Exotic Coarse Spaces for Domain Decomposition Methods 65
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M5 Schedule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66Martin J. Gander . . . . . . . . . . . . . . . . . . . . . . . . . . 67Clark Dohrmann . . . . . . . . . . . . . . . . . . . . . . . . . . . 68Joerg Willems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69Kevin Santugini . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
M6: Heterogeneous Domain Decomposition Methods 71M6P1 Schedule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
Heiko Berninger . . . . . . . . . . . . . . . . . . . . . . . . . . . 73Pablo Javier Blanco . . . . . . . . . . . . . . . . . . . . . . . . . 74Eva Casoni . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75Paola Gervasio . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
M6P2 Schedule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77Simona Perotto . . . . . . . . . . . . . . . . . . . . . . . . . . . 78Franz Rammerstorfer . . . . . . . . . . . . . . . . . . . . . . . . 79Human Rezaijafari . . . . . . . . . . . . . . . . . . . . . . . . . 80Anton Schiela . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
M7: Domain Decomposition, Preconditioning and Solvers in Isogeo-metric Analysis 83M7P1 Schedule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
Remi Abgrall . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85Michel Bercovier . . . . . . . . . . . . . . . . . . . . . . . . . . . 86Victor M. Calo . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87Krishan Gahalaut . . . . . . . . . . . . . . . . . . . . . . . . . . 88
M7P2 Schedule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89Christian Hesch . . . . . . . . . . . . . . . . . . . . . . . . . . . 90Stefan Kleiss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
M7P3 Schedule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92Angela Kunoth . . . . . . . . . . . . . . . . . . . . . . . . . . . 93Luca Franco Pavarino . . . . . . . . . . . . . . . . . . . . . . . . 94Satyendra Tomar . . . . . . . . . . . . . . . . . . . . . . . . . . 95Rafael Vazquez . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
M8: Domain Decomposition Techniques in Practical Flow Applica-tions 97M8 Schedule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
Eric Blayo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99Bas van ’t Hof . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100Mart Borsboom . . . . . . . . . . . . . . . . . . . . . . . . . . . 101Fred Wubs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
M9: Fast Solvers for Helmholtz and Maxwell equations 103M9P1 Schedule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
Lea Conen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105Hui Zhang . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
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Erwin Veneros . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107Bertrand Thierry . . . . . . . . . . . . . . . . . . . . . . . . . . 108
M9P2 Schedule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109Olaf Steinbach . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110Jin-Fa Lee . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111Eric Darrigrand . . . . . . . . . . . . . . . . . . . . . . . . . . . 112Yogi Ahmad Erlangga . . . . . . . . . . . . . . . . . . . . . . . . 113
M9P3 Schedule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114Rosalie Belanger-Rioux . . . . . . . . . . . . . . . . . . . . . . . 115Achim Schadle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116Ana Alonso Rodriguez . . . . . . . . . . . . . . . . . . . . . . . 117Martin Huber . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
M9P4 Schedule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119Ronan Perrussel . . . . . . . . . . . . . . . . . . . . . . . . . . . 120Jack Poulson . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121Zhen Peng . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122Stephane Lanteri . . . . . . . . . . . . . . . . . . . . . . . . . . 123
M10: New Developments of FETI, BDDC, and Related Domain De-composition Methods 125M10P1 Schedule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
Max Dryja . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127Juan Galvis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128Hyea Hyun Kim . . . . . . . . . . . . . . . . . . . . . . . . . . . 129Chang-Ock Lee . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
M10P2 Schedule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131Jungho Lee . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132Lourenco Beirao da Veiga . . . . . . . . . . . . . . . . . . . . . . 133Xuemin Tu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134Olof B. Widlund . . . . . . . . . . . . . . . . . . . . . . . . . . . 135Jun Zou . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
M11: Decomposition Strategies for Boltzmann’s Equation 137M11P1 Schedule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
Patrick Le Tallec . . . . . . . . . . . . . . . . . . . . . . . . . . 139Mohammed Lemou . . . . . . . . . . . . . . . . . . . . . . . . . 140Emmanuel Frenod . . . . . . . . . . . . . . . . . . . . . . . . . . 141Heiko Berninger . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
M11P2 Schedule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143Francois Golse . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144Giacomo Dimarco . . . . . . . . . . . . . . . . . . . . . . . . . . 145Sudarshan Tiwari . . . . . . . . . . . . . . . . . . . . . . . . . . 146Jerome Michaud . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
M12: Domain Decomposition Techniques in Life Science Modelingand Simulation 149
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M12 Schedule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150Oliver Rheinbach . . . . . . . . . . . . . . . . . . . . . . . . . . 151Simone Scacchi . . . . . . . . . . . . . . . . . . . . . . . . . . . 152Nejib Zemzemi . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153Gwenol Grandperrin . . . . . . . . . . . . . . . . . . . . . . . . 154
M13: Robust Multilevel Methods for Multiscale Problems 155M13P1 Schedule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
Victorita Dolean . . . . . . . . . . . . . . . . . . . . . . . . . . . 157Nicole Spillane . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158Jinchao Xu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159Juan Galvis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160Clark Dohrmann . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
M13P2 Schedule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162Petr Vanek . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163Robert Scheichl . . . . . . . . . . . . . . . . . . . . . . . . . . . 164James Brannick . . . . . . . . . . . . . . . . . . . . . . . . . . . 165Marco Buck . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
M13P3 Schedule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167Florian Thomines . . . . . . . . . . . . . . . . . . . . . . . . . . 168Ivan G. Graham . . . . . . . . . . . . . . . . . . . . . . . . . . . 169Jan Martin Nordbotten . . . . . . . . . . . . . . . . . . . . . . . 170Xiaozhe Hu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
M13P4 Schedule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172Baptiste Poirriez . . . . . . . . . . . . . . . . . . . . . . . . . . . 173Thomas Dufaud . . . . . . . . . . . . . . . . . . . . . . . . . . . 174Svetozar Margenov . . . . . . . . . . . . . . . . . . . . . . . . . 175Johannes Kraus . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
M14: 100% Parallelizable Algorithms for Symmetric, Indefinite andNon-Symmetric Problems 177M14 Schedule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
Ismael Herrera . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179Luis Miguel de la Cruz . . . . . . . . . . . . . . . . . . . . . . . 180Alberto Rosas . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181Ivan Contreras . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
M15: Space-Time Parallel Methods 183M15P1 Schedule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
Yvon Maday . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185Michael Minion . . . . . . . . . . . . . . . . . . . . . . . . . . . 186Rim Guetat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187Felix Kwok . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
M15P2 Schedule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189Stefan Guttel . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190Martin J. Gander . . . . . . . . . . . . . . . . . . . . . . . . . . 191
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Jacques Laskar . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192Julien Salomon . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
M15P3 Schedule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194Bankim Chandra Mandal . . . . . . . . . . . . . . . . . . . . . . 195Mohamed Kamel Riahi . . . . . . . . . . . . . . . . . . . . . . . 196Ronald Haynes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197Olga Mula Hernandez . . . . . . . . . . . . . . . . . . . . . . . . 198
M16: Domain Decomposition with Mortars 199M16P1 Schedule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
Frederic Hecht . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201Alfio Quarteroni . . . . . . . . . . . . . . . . . . . . . . . . . . . 202Caroline Japhet . . . . . . . . . . . . . . . . . . . . . . . . . . . 203Zakaria Belhachmi . . . . . . . . . . . . . . . . . . . . . . . . . . 204
M16P2 Schedule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205Christian Waluga . . . . . . . . . . . . . . . . . . . . . . . . . . 206Yvon Maday . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207Francois-Xavier Roux . . . . . . . . . . . . . . . . . . . . . . . . 208Oldrich Vlach . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209Todd Arbogast . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210
M17: Domain Decomposition Methods based on Robin Conditionsfor Large and / or Nonlinear Problems 211M17P1 Schedule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212
Sebastien Loisel . . . . . . . . . . . . . . . . . . . . . . . . . . . 213Florence Hubert . . . . . . . . . . . . . . . . . . . . . . . . . . . 214Oliver Sander . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215Minh-Binh Tran . . . . . . . . . . . . . . . . . . . . . . . . . . . 216
M17P2 Schedule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217Soheil Hajian . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218Ronald Haynes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219Joel Phillips . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220Yingxiang Xu . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
M18: Solvers for Discontinuous Galerkin Methods 223M18P1 Schedule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224
Alexandre B. Pieri . . . . . . . . . . . . . . . . . . . . . . . . . . 225Kolja Brix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226Christoph Lehrenfeld . . . . . . . . . . . . . . . . . . . . . . . . 227Eun-Hee Park . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228
M18P2 Schedule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229Paola F. Antonietti . . . . . . . . . . . . . . . . . . . . . . . . . 230Andrew T. Barker . . . . . . . . . . . . . . . . . . . . . . . . . . 231Guido Kanschat . . . . . . . . . . . . . . . . . . . . . . . . . . . 232Ludmil Zikatanov . . . . . . . . . . . . . . . . . . . . . . . . . . 233
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M19: Domain Decomposition in Computational Cardiology 235M19P1 Schedule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236
Luca Gerardo-Giorda . . . . . . . . . . . . . . . . . . . . . . . . 237Dorian Krause . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238Stefano Zampini . . . . . . . . . . . . . . . . . . . . . . . . . . . 239Charles Pierre . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240
M19P2 Schedule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241Martin Weiser . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242Gernot Plank . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243Ricardo Ruiz Baier . . . . . . . . . . . . . . . . . . . . . . . . . 244Maxime Sermesant . . . . . . . . . . . . . . . . . . . . . . . . . 245
M20: Domain Decomposition and Multiscale Methods 247M20 Schedule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248
Talal Rahman . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249Juan Galvis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250Robert Scheichl . . . . . . . . . . . . . . . . . . . . . . . . . . . 251Rui Du . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252
C1: Contact and Mechanics Problems 253Jaroslav Haslinger . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254Brahim Nouiri . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255Ihor I. Prokopyshyn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256Alexandros Markopoulos . . . . . . . . . . . . . . . . . . . . . . . . . . 257
C2: Contact and Mechanics Problems 259Daniel Choi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260Vincent Visseq . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261Geoffrey Desmeure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262Julien Riton . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263Philippe Karamian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264
C3: Optimized Schwarz Methods 265Florence Hubert . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266Lahcen Laayouni . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267Erell Jamelot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268Frederic Magoules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269
C4: Domain Decomposition for Helmholtz Equation 271Chris Stolk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272Dalibor Lukas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273
C5: Heterogeneous Problems and Coupling Methods 275Jonathan Youett . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276Manel Tayachi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277
C6: Heterogeneous Problems and Coupling Methods 279
xiv
Marco Discacciati . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280Marina Vidrascu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281Christian Engwer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282
C7: Domain Decomposition with Preconditionners 283Daniel Szyld . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284Feng-Nan Hwang . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285Santiago Badia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286Laurent Berenguer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288
C8: Application to Flow Problems 289Aivars Zemitis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290Leonardo Baffico . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291Francois Pacull . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292Daniel Loghin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293
C9: Multidomains and Time Domain Decomposition 295Chao Yang . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296Felix Kwok . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297Martin Cermak . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298Damien Tromeur-Dervout . . . . . . . . . . . . . . . . . . . . . . . . . . 299
C10: Application to Flow Problems 301Jyri Leskinen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302Mohamed Khaled Gdoura . . . . . . . . . . . . . . . . . . . . . . . . . . 303Thu Huyen Dao . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304Guillaume Houzeaux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305
C11: Time Parallel - Parareal Methods 307Noha Makhoul-Karam . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308Daniel Ruprecht . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309Rolf Krause . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310
C12: Optimization Methods/Probabilistic Methods 311Andreas Langer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312Firmim Andzembe Okoubi . . . . . . . . . . . . . . . . . . . . . . . . . 313Francisco Bernal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314Samia Riaz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315
C13: FETI Methods 317Leszek Marcinkowski . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318Ange Toulougoussou . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319Hui Zhang . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 320Christian Rey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321
C14: Time Dependent PDEs and Applications 323Petros Aristidou . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324
xv
Rodrigue Kammogne . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325Frederic Plumier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326David Cherel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327
C15: Multigrid Methods 329Kab Seok Kang . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 330Pawan Kumar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331Lori Badea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332
C16: Finite Element Method for Domain Decomposition 333Patrick Le Tallec . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334Thomas Dickopf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335Debasish Pradhan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336Frederic Magoules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337
C17: Non-matching Grids/Nonconforming Discretization 339Kirill Pichon Gostaf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 340Ajit Patel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341Eliseo Chacon Vera . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342Beatriz Eguzkitza . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343
C18: FETI Methods 345K. C. Park . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346Ulrich Langer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347
C19: Multiprocessors Applications 349Hatem Ltaief . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350Menno Genseberger . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351
C20: Adaptive Meshing Paradigm 353Shuo Zhang . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354Cedric Lachat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355
C21: FETI Methods 357Marta Jarosova . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 358Michal Merta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359
Author Index 361
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Schedule
Time Event Location
Monday, June 257:45-8:45 Registration INRIA Reception8:45-9:15 Opening Remarks Amphi
9:15-10:00 Plenary (P1 - Laurence Halpern) Amphi10:00-10:30 Coffee Break10:30-12:15 Parallel Sessions (M16P1, M10P1, M14, M6P1, M5, C1) Amphi, Markov, Petri, Turing, I50, I5112:15-14:00 Lunch14:00-14:45 Plenary (P2 - Geraldine Pichot) Amphi14:45-15:30 Plenary (P3 - Axel Klawonn) Amphi15:30-16:00 Coffee Break16:00-18:10 Parallel Sessions (M16P2, M10P2, M13P1, M6P2, M8, C2) Amphi, Markov, Petri, Turing, I50, I5119:00-20:00 Welcoming cocktail
Tuesday, June 268:30-9:15 Plenary (P4 - Marcus Sarkis) Amphi
9:15-10:00 Plenary (P5 - Jin-Fa Lee) Amphi10:00-10:30 Coffee Break10:30-12:15 Parallel Sessions (M7P1, M15P1, M13P2, M9P1, M11P1, C6/C18) Amphi, Markov, Petri, Turing, I50, I5112:15-14:00 Lunch14:00-14:45 Plenary (P6 - Clemens Pechstein), Amphi14:45-15:35 Parallel Sessions (M7P2, C4, C19, C5, C20, C21) Amphi, Markov, Petri, Turing, I50, I5115:35-16:00 Coffee Break16:00-17:45 Parallel Sessions (M7P3, M15P2, M13P3, M9P2, M11P2, C7) Amphi, Markov, Petri, Turing, I50, I5118:00-22:00 Scientific Committee Meeting
Wednesday, June 278:30-9:15 Plenary (P7 - Hyea Hyun Kim) Amphi
9:15-10:00 Plenary (P8 - Beatrice Riviere) Amphi10:00-10:30 Coffee Break10:30-12:15 Parallel Sessions (M2P1, M15P3, M13P4, M9P3, C3, C8) Amphi, Markov, Petri, Turing, I50, I5112:15-14:00 Lunch14:00-14:45 Plenary (P9 - Xiao-Chuan Cai) Amphi14:45-15:30 Plenary (P10 - Eberhard Bansch) Amphi15:30-16:00 Coffee Break16:00-17:45 Parallel Sessions (M2P2, M18P1, M20, M9P4, C9, C10) Amphi, Markov, Petri, Turing, I50, I51
Thursday, June 288:30-9:15 Plenary (P11 - Blanca Ayuso de Dios) Amphi
9:15-10:00 Plenary (P12 - Chen-Song Zhang) Amphi10:00-10:30 Coffee Break10:30-12:15 Parallel Sessions (M17P1, M18P2, M19P1, M3, C11/C18, C14) Amphi, Markov, Petri, Turing, I50, I5112:15-13:30 Lunch13:30-22:00 Excursion and dinner
Friday, June 298:30-9:15 Plenary (P13 - Ralf Hiptmair) Amphi
9:15-10:00 Plenary (P14 - Michael Holst) Amphi10:00-10:30 Coffee Break10:30-12:15 Parallel Sessions (M17P2, M1, M19P2, M4, M12, C12) Amphi, Markov, Petri, Turing, I50, I5112:15-14:00 Lunch14:00-15:45 Parallel Sessions (C15, C13, C16, C17) Amphi, Markov, Petri, Turing15:45-16:15 Closing
1
M T W T F Monday, June 25, 2012
7:45-8:45 Registration
8:45-9:15 Opening
Plenary P1 (Chair: Ralf Kornhuber)
9:15-10:00 Laurence Halpern
10:00-10:30 Coffee Break
M16 P1 M10 P1 M14 M6 P1 M5 C1
10:30-12:15 Amphi Markov Petri Turing I50 I51
Frederic Hecht Maksymilian Dryja Ismael Herrera Heiko Berninger Martin J. Gander Jaroslav Haslinger
Alfio Quarteroni Juan Galvis Luis Miguel de la Cruz Pablo Javier Blanco Clark Dohrmann Brahim Nouiri
Caroline Japhet Hyea Hyun Kim Alberto Rosas Eva Casoni Jorg Willems Ihor I. Prokopyshyn
Zakaria Belhachmi Chang-Ock Lee Ivan Contreras Paola Gervasio Kevin Santugini Alexandros Markopoulos
12:15-14:00 Lunch
Plenary P2 (Chair: Ralf Kornhuber)
14:00-14:45 Geraldine Pichot
Plenary P3 (Chair: Ralf Kornhuber)
14:45-15:30 Axel Klawonn
15:30-16:00 Coffee Break
M16 P2 M10 P2 M13 P1 M6 P2 M8 C2
16:00-18:10 Amphi Markov Petri Turing I50 I51
Christian Waluga Jungho Lee Victorita Dolean Simona Perotto Eric Blayo Daniel Choi
Yvon Maday L. Beirao da Veiga Nicole Spillane Franz Rammerstorfer Bas van ’t Hof Vincent Visseq
Francois-Xavier Roux Xuemin Tu Jinchao Xu Human Rezaijafari Mart Borsboom Geoffrey Desmeure
Oldrich Vlach Olof Widlund Juan Galvis Anton Schiela Fred Wubs Julien Riton
Todd Arbogast Jun Zou Clark Dohrmann Philippe Karamian
19:00-20:00 Welcoming cocktail
2
M T W T F Tuesday, June 26, 2012
Plenary P4 (Chair: Alfio Quarteroni)
8:30-9:15 Marcus Sarkis
Plenary P5 (Chair: Alfio Quarteroni)
9:15-10:00 Jin-Fa Lee
10:00-10:30 Coffee Break
M7 P1 M15 P1 M13 P2 M9 P1 M11 P1 C6/C18
10:30-12:15 Amphi Markov Petri Turing I50 I51
Remi Abgrall Yvon Maday Petr Vanek Lea Conen Patrick Le Tallec Marco Discacciati
Michel Bercovier Michael Minion Robert Scheichl Hui Zhang Mohammed Lemou Marina Vidrascu
Victor M. Calo Rim Guetat James Brannick Erwin Veneros Emmanuel Frenod Christian Engwer
Krishan P. S. Gahalaut Felix Kwok Marco Buck Bertrand Thierry Heiko Berninger K. C. Park (C18)
12:15-14:00 Lunch
Plenary P6 (Chair: David Keyes)
14:00-14:45 Clemens Pechstein
M7 P2 C4 C19 C5 C20 C21
14:45-15:35 Amphi Markov Petri Turing I50 I51
Christian Hesch Chris Stolk Hatem Ltaief Jonathan Youett Shuo Zhang Marta Jarosova
Stefan Kleiss Dalibor Lukas Menno Genseberger Manel Tayachi Cedric Lachat Michal Merta
15:35-16:00 Coffee Break
M7 P3 M15 P2 M13 P3 M9 P2 M11 P2 C7
16:00-17:45 Amphi Markov Petri Turing I50 I51
Angela Kunoth Stefan Guttel Florian Thomines Olaf Steinbach Francois Golse Daniel Szyld
Luca F. Pavarino Martin J. Gander Ivan Graham Jin-Fa Lee Giacomo Dimarco Feng-Nan Hwang
Satyendra Tomar Jacques Laskar Jan Nordbotten Eric Darrigrand Sudarshan Tiwari Santiago Badia
Rafael Vazquez Julien Salomon Xiaozhe Hu Yogi Erlangga Jerome Michaud Laurent Berenguer
18:00-22:00 Scientific committee meeting
3
M T W T F Wednesday, June 27, 2012
Plenary P7 (Chair: Laurence Halpern)
8:30-9:15 Hyea Hyun Kim
Plenary P8 (Chair: Laurence Halpern)
9:15-10:00 Beatrice Riviere
10:00-10:30 Coffee Break
M2 P1 M15 P3 M13 P4 M9 P3 C3 C8
10:30-12:15 Amphi Markov Petri Turing I50 I51
Oliver Sander Bankim Mandal Baptiste Poirriez Rosalie Belanger-Rioux Florence Hubert Aivars Zemitis
Thi Thao Phuong Hoang Mohamed Kamel Riahi Thomas Dufaud Achim Schadle Lahcen Laayouni Leonardo Baffico
Frederic Nataf Ron Haynes Svetozar Margenov Ana Alonso Rodriguez Erell Jamelot Francois Pacull
Zhangxin Chen Olga Mula Hernandez Johannes Kraus Martin Huber Frederic Magoules Daniel Loghin
12:15-14:00 Lunch
Plenary P9 (Chair: Petter Bjørstad)
14:00-14:45 Xiao-Chuan Cai
Plenary P10 (Chair: Petter Bjørstad)
14:45-15:30 Eberhard Bansch
15:30-16:00 Coffee Break
M2 P2 M18 P1 M20 M9 P4 C9 C10
16:00-17:45 Amphi Markov Petri Turing I50 I51
Bernd Flemisch Alexandre Pieri Talal Rahman Ronan Perrussel Chao Yang Jyri Leskinen
Paul-Marie Berthe Kolja Brix Juan Galvis Jack Poulson Felix Kwok M. Khaled Gdoura
J.-B. Apoung Kamga Christoph Lehrenfeld Robert Scheichl Zhen Peng Martin Cermak Thu Huyen Dao
Anthony Michel Eun-Hee Park Rui Du Stephane Lanteri D. Tromeur-Dervout Guillaume Houzeaux
4
M T W T F Thursday, June 28, 2012
Plenary P11 (Chair: Olof Widlund)
8:30-9:15 Blanca Ayuso de Dios
Plenary P12 (Chair: Olof Widlund)
9:15-10:00 Chen-Song Zhang
10:00-10:30 Coffee Break
M17 P1 M18 P2 M19 P1 M3 C11/C18 C14
10:30-12:15 Amphi Markov Petri Turing I50 I51
Sebastien Loisel Paola F. Antonietti Luca Gerardo-Giorda James Adler Noha Makhoul-Karam Petros Aristidou
Florence Hubert Andrew Barker Dorian Krause Pavel Bochev Daniel Ruprecht Rodrigue Kammogne
Oliver Sander Guido Kanschat Stefano Zampini Fleurianne Bertrand Rolf Krause Frederic Plumier
Minh Binh Tran Ludmil T. Zikatanov Charles Pierre Steffen Munzenmaier Ulrich Langer (C18) David Cherel
12:15-13:30 Lunch
13:30-22:00 Excursion and dinner
5
M T W T F Friday, June 29, 2012
Plenary P13 (Chair: Susanne Brenner)
8:30-9:15 Ralf Hiptmair
Plenary P14 (Chair: Susanne Brenner)
9:15-10:00 Michael Holst
10:00-10:30 Coffee Break
M17 P2 M1 M19 P2 M4 M12 C12
10:30-12:15 Amphi Markov Petri Turing I50 I51
Soheil Hajian Frederic Hecht Martin Weiser Martin J. Gander Oliver Rheinbach Andreas Langer
Ronald Haynes Pierre Jolivet Gernot Plank Xuemin Tu Simone Scacchi Firmim Andzembe Okoubi
Joel Phillips Christophe Prud’homme Ricardo Ruiz Baier Francois-Xavier Roux Nejib Zemzemi Francisco Bernal
Yingxiang Xu Abdoulaye Samake Maxime Sermesant Olof Widlund Gwenol Grandperrin Samia Riaz
12:15-14:00 Lunch
C15 C13 C16 C17
14:00-15:45 Amphi Markov Petri Turing
Kab Seok Kang Leszek Marcinkowski Patrick Le Tallec Kirill Pichon Gostaf
Pawan Kumar Ange Toulougoussou Thomas Dickopf Ajit Patel
Lori Badea Hui Zhang Debasish Pradhan Eliseo Chacon Vera
Christian Rey Frederic Magoules Beatriz Eguzkitza
15:45-16:15 Closing
6
Plenary Lecture P1Schedule Author Index Session Index
Date: Monday, June 25Time: 9:15-10:00Location: AmphiChairman: Ralf Kornhuber
9:15-10:00 : Laurence HalpernOptimized Schwarz Waveform Relaxation and Applications to SemilinearEquationsAbstract
7
Optimized Schwarz Waveform Relaxation andApplications to Semilinear EquationsSchedule Author Index Session Index
Laurence HalpernLAGA Universite Paris [email protected]
AbstractOptimized Schwarz waveform relaxation have been presented for the first time inDD11 in 1999 [2]. These algorithms permit to solve the equations in time win-dows,in different subdomains in space, exchanging informations at the end of thetime interval on the interfaces. They use Robin or Ventcell transmission conditions,and can be used without overlap if necessary. A key issue in this process is to choosethe coefficients in order to accelerate the convergence.We concentrate in this presentation on parabolic equations and systems. The coef-ficients of the transmission operators can be characterized through a best approxi-mation problem on a compact set whose dimensions are related to the parametersof the discretization. For the advection diffusion equation, a complete analysis inone dimension was performed in [1], some results were announced in DD19, butcomplete general formulas will only be available for DD21. The first part of my talkwill deal with this important question.As soon as this is solved, we can address the extension to nonlinear problems, withvarious questions in mind1) How to use the optimized coefficients ?2) How to prove convergence of the algorithm ?3) How to relate the Schwarz waveform relaxation algorithm and the algorithm ofresolution of nonlinear problem (like Newton’s algorithm).All these questions will be considered, and applications to the reactive transportsystem will be presented.
[1] D. Bennequin, M. Gander and L. Halpern. A Homographic Best Approxima-tion Problem with Application to Optimized Schwarz Waveform Relaxation. Math.Comp. 78 (2009), no. 265, 185223.[2] M. Gander, L. Halpern and F. Nataf. Optimal Convergence for Overlapping andNon-Overlapping Schwarz Waveform Relaxation. Eleventh International Confer-ence on Domain Decomposition Methods (London, 1998), 2736 (electronic), DDM.org,Augsburg, 1999.
8
Plenary Lecture P2Schedule Author Index Session Index
Date: Monday, June 25Time: 14:00-14:45Location: AmphiChairman: Ralf Kornhuber
14:00-14:45 : Geraldine PichotOn Robust Numerical Methods for Solving Flow in Stochastic FractureNetworksAbstract
9
On Robust Numerical Methods for Solving Flow inStochastic Fracture NetworksSchedule Author Index Session Index
Geraldine PichotINRIA, Rennes Bretagne Atlantique, [email protected]
AbstractWorking with random domains requires the development of specific and robust nu-merical methods to be able to solve physical phenomena whatever the generatedgeometries. Hydrogeology is a typical area of application where one has to face un-certainty about the geometry and the properties of the domain since the availableinformation on the underground media is local, gathered through in-situ experi-ments with outcrops and wells. From measurements, statistical laws are derivedthat allow the generation of natural-like random media.The focus of this talk will concern flow in discrete fracture networks. The parame-ters governing the fractures lengths, shapes, orientations, positions as well as theirhydraulic conductivity are stochastic. Our objective is to design robust numericalmethods to solve Poiseuille’s flow in large and heterogeneous stochastic fracturenetworks.The first part will deal with the meshing strategies required to obtain a good qualitymesh for any generated networks. The second part will be devoted to numericaltechniques to solve the flow equations. A Mortar-like method to deal with non-conforming meshes at the fracture intersections will be presented as well as a Schurcomplement approach to solve the linear system of interest in parallel.This work is a joined work with Jocelyne Erhel, Baptiste Poirriez, Jean-Raynald deDreuzy, Patrick Laug and Thomas Dufaud.
10
Plenary Lecture P3Schedule Author Index Session Index
Date: Monday, June 25Time: 14:45-15:30Location: AmphiChairman: Ralf Kornhuber
14:45-15:30 : Axel KlawonnDeflation, Projector Preconditioning and Robust Domain DecompositionMethodsAbstract
11
Deflation, Projector Preconditioning and RobustDomain Decomposition MethodsSchedule Author Index Session Index
Axel KlawonnUniversitat zu Koln, Mathematisches [email protected]
AbstractIn this talk, projector preconditioning, also known as the deflation method, is ap-plied to the FETI-DP and the BDDC method in order to create a second, indepen-dent coarse problem. It may as well be used to improve the robustness, e.g., foralmost incompressible elasticity problems and second order elliptic partial differ-ential equations with discontinuous coefficients. In addition, it will be shown thatstandard FETI-DP methods are robust for elasticity problems with respect to coef-ficient jumps within the subdomains. Here, the convergence of FETI-DP methodsfor problems in 3D with almost incompressible inclusions or compressible inclusionswith different material parameters embedded in a compressible matrix material isanalyzed. It can also be demonstrated that these FETI-DP algorithms are robustfor challenging problems from nonlinear biomechanics with almost incompressiblematerial properties. Finally, it will be reported on results for certain overlappingSchwarz methods for elliptic problems with discontinuous coefficents not alignedwith the interface.The results in this talk are based on different joint projects with Sabrina Gippert,Patrick Radtke, Oliver Rheinbach, and Olof Widlund.
12
Plenary Lecture P4Schedule Author Index Session Index
Date: Thursday, June 28Time: 8:30-9:15Location: AmphiChairman: Alfio Quarteroni
8:30-9:15 : Marcus SarkisDDMs for DG DiscretizationsAbstract
13
DDMs for DG DiscretizationsSchedule Author Index Session Index
Marcus SarkisWorcester Polytechnic Institute (WPI/USA and IMPA/Brazil)[email protected]
AbstractWe consider a second order elliptic equation with discontinuous coefficients. Thedomain is defined as a geometrically (possibly nonconforming) decomposition ofsubstructures. Inside each substructure, a conforming triangulation is introducedand a conforming or DG finite element method is considered. To handle nonmatch-ing meshes and coefficient jumps across substructure interfaces, we consider properDG discretizations. The first part of the talk we discuss a priori error estimatesincluding the case where the coefficient is anisotropic. The second part of the talkwe discuss solvers based on FETI-DP, BDDC, Neumann-Neumann and AverageSchwarz methods which are robust with respect to coefficients jumps, number ofsubdomains, local mesh sizes and mesh sizes ratio across substructure interfaces.Cases where the coefficient varies inside the substructures are also discussed. Nu-merical results are presented.The results were obtained in collaboration with Prof. Maksymilian Dryja, Dr. JuanGalvis and Dr. Piotr Krzyzanowski.
14
Plenary Lecture P5Schedule Author Index Session Index
Date: Tuesday, June 26Time: 9:15-10:00Location: AmphiChairman: Alfio Quarteroni
9:15-10:00 : Jin-Fa LeeAn Expedition to Solving a Multiscale Electromagnetic ProblemAbstract
15
An Expedition to Solving a Multiscale ElectromagneticProblemSchedule Author Index Session Index
Jin-Fa LeeESL, ECE Dept., The Ohio State [email protected]
AbstractThis talk centers on the full-wave solution of an electromagnetic wave scatteringfrom a composite aircraft with multi-scale geometrical features. As in many en-gineering applications, the target/object considered herein is built by putting to-gether many different parts and components, and each of them can be changedand modified due to the design and operational needs. For example, the mockupfighter jet can be carrying different pay loads for different missions, varying thethickness of the lossy thin coatings to study the effectiveness of radar absorption,and fine tuning the engine inlet to reduce EM echo area etc. As a consequence,a full-wave solution strategy which incorporates hierarchical geometrical partition-ing/decomposition seamlessly would be highly desirable for many mission criticalengineering studies. In solving the EM wave scattering from such a complicated andmultiscale composite aircraft, we have encountered many unexpected surprises aswell as expected technical difficulties. At the end, our pursuit of a rigorous full-wavesolution proved to be prolific, thought-provoking, but most of them all, educational.In this talk, I shall elucidate a few major highlights of our journey:
• Non-conformal integral equation domain decomposition methods for multi-scale EM problems.
• A generalized combined field integral equation method with multiple surfacetraces for modeling penetrable targets.
• An integral equation domain decomposition method for EM wave scatteringfrom deep cavities such as engine inlets.
• A multiscale finite element method to incorporate honeycomb metamaterialsstructures on a dielectric radome.
• The use of the best local sub-domain computational electromagnetic (CEM)solver and the multi-solver domain decomposition method.
16
Plenary Lecture P6Schedule Author Index Session Index
Date: Tuesday, June 26Time: 14:00-14:45Location: AmphiChairman: David Keyes
14:00-14:45 : Clemens PechsteinSubstructuring for Multiscale ProblemsAbstract
17
Substructuring for Multiscale ProblemsSchedule Author Index Session Index
Clemens PechsteinJohannes Kepler University, Linz, [email protected]
AbstractFETI, FETI-DP, and the related balancing Neumann-Neumann and BDDC meth-ods are the most widely used iterative substructuring methods for the solutionof large sparse systems stemming from finite element discretizations of ellipticpartial differential equations. For simplicity, consider the scalar elliptic equation−div(α∇u) = f . However, we let the diffusion coefficient α vary over many ordersof magnitude in an unstructured way on the computational domain, which justifiesto call this a multiscale problem. It is known that if α is resolved by the subdomainpartitioning (i.e. constant in each subdomain), then the FETI and FETI-DP pre-conditioners (and their balancing counterparts) can be made robust with respectto the jumps of α across subdomain interfaces. However, for many highly varyingcoefficients, a straightforward application of the ’standard’ delivers pessimistic con-dition number bounds. In the first part of this talk, I will discuss the application(and adaption) of FETI methods to the multiscale problem above. Using weightedPoincare inequalities – a theoretical tool that is interesting in itself – robustness ofFETI can be proved rigorously under certain monotonicity conditions on α. We willalso investigate in how far these conditions are necessary. Furthermore, for piece-wise constant coefficients α, the performance of FETI depends on the ”geometry”of α (i.e. on the subregions where α is constant), and I will work out how to quantifythis dependence. The second part of this talk is devoted to the more difficult case ofFETI-DP, which requires weighted Poincare inequalities with suitably weighted av-erages. Finally, for real-life problems with highly varying coefficients, the challengeis how to adapt the constraints in FETI-DP/BDDC in order to achieve robustness,and I will sketch some ideas into this direction. The research underlying this talkwas done in joint collaboration with Rob Scheichl, Marcus Sarkis, and also withClark Dohrmann.
18
Plenary Lecture P7Schedule Author Index Session Index
Date: Wednesday, June 27Time: 8:30-9:15Location: AmphiChairman: Laurence Halpern
8:30-9:15 : Hyea Hyun KimRecent Advances in Domain Decomposition Methods for the Stokes ProblemAbstract
19
Recent Advances in Domain Decomposition Methodsfor the Stokes ProblemSchedule Author Index Session Index
Hyea Hyun KimDepartment of Applied Mathematics, Kyung Hee University, [email protected]
Chang-Ock LeeDepartment of Mathematical Sciences, KAIST, [email protected]
Eun-Hee ParkCenter for Computation and Technology, Louisiana State University, [email protected]
AbstractDomain decomposition methods for the Stokes problem are developed under a moregeneral framework, which allows both continuous and discontinuous pressure func-tions and more flexibility in the construction of the coarse problem. For the case ofdiscontinuous pressure functions, a coarse problem related to only primal velocityunknowns is shown to give scalability in both dual and primal types of domaindecomposition methods. The two formulations are shown to have the same extremeeigenvalues and the ratio of the two extreme eigenvalues weakly depends on the lo-cal problem size. This property results in a good scalability in both the primal anddual formulations for the case with discontinuous pressure functions. The primalformulation can also be applied to the case with continuous pressure functions andvarious numerical experiments are carried out to present promising features of ourapproach.
20
Plenary Lecture P8Schedule Author Index Session Index
Date: Wednesday, June 27Time: 9:15-10:00Location: AmphiChairman: Laurence Halpern
9:15-10:00 : Beatrice RiviereDiscontinuous Galerkin Methods for Multiphysics ProblemsAbstract
21
Discontinuous Galerkin Methods for MultiphysicsProblemsSchedule Author Index Session Index
Beatrice RiviereRice [email protected]
AbstractThe numerical solution of coupled subdomains characterized by different types offlows is presented. Examples of such coupled flows include the environmental prob-lem of groundwater contamination through rivers or the industrial manufacturing offilters. The coupling at the interfaces between subdomains is based on the Beavers-Joseph-Saffman conditions.The proposed algorithms employ discontinuous Galerkin methods. These meth-ods are well-suited to coupling different physics at different scales. Information istransmitted through fluxes defined on the interface. In addition, the use of adap-tive mesh refinement and non-matching grids is facilitated by the lack of continuityconstraints between the mesh elements. A monolithic approach is compared with adecoupled approach based on the two-grid technique. Finally, the coupling of dis-continuous Galerkin methods with finite element methods or finite volume methodsis formulated in a domain decomposition setting.
22
Plenary Lecture P9Schedule Author Index Session Index
Date: Wednesday, June 27Time: 14:00-14:45Location: AmphiChairman: Petter Bjørstad
14:00-14:45 : Xiao-Chuan CaiMonolithic Schwarz Algorithms for Simulation and Optimization of BloodFlowsAbstract
23
Monolithic Schwarz Algorithms for Simulation andOptimization of Blood FlowsSchedule Author Index Session Index
Xiao-Chuan CaiDepartment of Computer Science, University of Colorado at Boulder, Boulder, CO80309, [email protected]
AbstractThe class of overlapping Schwarz algorithms has been well studied for elliptic prob-lems. In this talk, we discuss the application of Schwarz algorithms for several morechallenging problems including the implicit solution of coupled fluid-structure in-teraction problems arising in the simulation of blood flows in compliant arteries andthe shape optimization of steady state incompressible flows. We show by numericalexperiments that, after some proper modifications, multilevel Schwarz algorithmswork quite well for these nonlinear systems of coupled multi-physics problems andgood scalability results are obtained on parallel machines with thousands of proces-sors. This is a joint work with Yuqi Wu and Rongliang Chen.
24
Plenary Lecture P10Schedule Author Index Session Index
Date: Wednesday, June 27Time: 14:45-15:30Location: AmphiChairman: Petter Bjørstad
14:45-15:30 : Eberhard BanschA Finite Element Method for Particulate FlowAbstract
25
A Finite Element Method for Particulate FlowSchedule Author Index Session Index
Eberhard BanschAM IIIDepartment MathematicsFriedrich-Alexander University [email protected]
Rodolphe PrignitzAM IIIDepartment MathematicsFriedrich-Alexander University [email protected]
AbstractParticulate flow, i.e. flow of a (Newtonian) carrier liquid loaded with rigid particles,plays an important role in many technical applications. From a mathematical pointof view, particulate flows give rise to an interesting and involved free boundaryproblem, where the flow field and the motion of the particles are coupled throughthe forces exerted by the flow and Newton’s law for the particles’ motion. In thistalk a one-domain finite element method to solve this problem in 2d and 3d ispresented. The main ingredients consist of a splitting scheme in time, a subspaceprojection method to account for the restriction of the flow field to a rigid bodymotion in those parts of the domain occupied by the particles and adaptivity toresolve the geometric problems.
26
Plenary Lecture P11Schedule Author Index Session Index
Date: Thursday, June 28Time: 8:30-9:15Location: AmphiChairman: Olof Widlund
8:30-9:15 : Blanca Ayuso de DiosSolvers for Discontinuous Galerkin MethodsAbstract
27
Solvers for Discontinuous Galerkin MethodsSchedule Author Index Session Index
Blanca Ayuso de DiosCentre de Recerca Matematica, Campus de Bellaterra, 08193 Bellaterra,Barcelona, [email protected]
AbstractThe talk will discuss the use of old and the design of new Subspace Correction andDomain Decomposition techniques for developing and analyzing efficient solvers forDG methods (for some very simple model problems).
28
Plenary Lecture P12Schedule Author Index Session Index
Date: Thursday, June 28Time: 9:15-10:00Location: AmphiChairman: Olof Widlund
9:15-10:00 : Chen-Song ZhangFast Auxiliary Space Preconditioning: Implementation and Applications inComplex FlowsAbstract
29
Fast Auxiliary Space Preconditioning:Implementation and Applications in Complex FlowsSchedule Author Index Session Index
Chen-Song ZhangChinese Academy of [email protected]
AbstractOver the last few decades, intensive research has been done on developing efficientand practical iterative solvers for discretized PDEs. One useful mathematical tech-nique, that has drawn a lot of attention recently, is a general framework calledAuxiliary Space Preconditioning. This framework represents a large class of meth-ods that transform a complicated system, by using auxiliary spaces, into a sequenceof simpler systems and construct efficient preconditioners with efficient solvers forthese simpler systems. In this talk, we will discuss recent development of thismethod for simulating simple and complex multiphase fluids. We will also intro-duce a new software library, FASP, designed and implemented for auxiliary spacepreconditioners. In particular, we will demonstrate some industrial applications ofthe FASP package for enhanced oil recovery techniques.
30
Plenary Lecture P13Schedule Author Index Session Index
Date: Friday, June 29Time: 8:30-9:15Location: AmphiChairman: Susanne Brenner
8:30-9:15 : Ralf HiptmairNovel Multi-Trace Boundary Element Methods for ScatteringAbstract
31
Novel Multi-Trace Boundary Element Methods forScatteringSchedule Author Index Session Index
Ralf HiptmairSAM, D-MATH, ETH Zurich, [email protected]
Xavier ClaeysISAE Toulouse, [email protected]
Carlos Jerez-HanckesSchool of Engineering, Pontificia Universidad Catolica de [email protected]
AbstractWe consider the scattering of acoustic or electromagnetic waves at a penetrable ob-ject composed of different homogeneous materials, that is, the material coefficientsare supposed to be piecewise constant in sub-domains. This makes possible to recastthe problem into boundary integral equations posed on the interfaces. Those can bediscretized by means of boundary elements (BEM). This approach is widely used innumerical simulations and often relies on so-called first-kind single-trace BIE, alsoknown as PMCHWT scheme in electromagnetics. These integral equations directlyarise from Calderon identities, but after BEM discretization give rise to poorly con-ditioned linear systems, for which no preconditioner seems to be available so far.
As a remedy we propose new multi-trace boundary integral equations; whereasthe single-trace BIE feature unique Cauchy traces on sub-domain interfaces as un-knowns, the multi-trace idea takes the cue from domain decomposition and tearsthe unknowns apart so that local Cauchy traces are recovered. Two of them live oneach interface and thus we dub the methods “multi-trace”. The benefit of localiza-tion is the possibility of Calderon preconditioning.
Multi-trace formulations come in two flavors. A first variant, the global multi-trace approach, is obtained from the single-trace equations by taking a “vanishinggap limit”, see [X. Claeys and R. Hiptmair, Boundary integral formulation ofthe first kind for acoustic scattering by composite structures, Comm. Pure Ap-plied Math., in press (2012)] and [X. Claeys and R. Hiptmair, Electromagneticscattering at composite objects: A novel multi-trace boundary integral formulation,M2AN, in press (2012)]. The second variant is the local multi-trace method andis based on local coupling across sub-domain interfaces, see [R. Hiptmair andC. Jerez-Hanckes, Multiple traces boundary integral formulation for Helmholtztransmission problems, Adv. Appl. Math., (2011), doi: 10.1007/s10444-011-9194-3]. Both methods are amenable to Calderon preconditioning.
32
Plenary Lecture P14Schedule Author Index Session Index
Date: Friday, June 29Time: 9:15-10:00Location: AmphiChairman: Susanne Brenner
9:15-10:00 : Michael HolstError Estimates and the Finite Element Exterior Calculus for CriticalExponent Problems in Geometric Analysis and General RelativityAbstract
33
Error Estimates and the Finite Element ExteriorCalculus for Critical Exponent Problems in GeometricAnalysis and General RelativitySchedule Author Index Session Index
Michael HolstDepartments of Mathematics and PhysicsUniversity of California, San [email protected]
AbstractWe consider adaptive methods for nonlinear critical exponent partial differentialequations arising in geometric analysis and mathematical physics. After present-ing some motivating examples, we describe an approach to establishing a prioriGalerkin finite element error estimates without the need for angle conditions tofirst obtain discrete pointwise control of the nonlinearity. We then show how thea priori error estimates can themselves be used to establish pointwise control ofdiscrete solutions, without the need for a discrete maximum principle, and henceagain without the need for angle conditions. We then describe a new approach toanalyzing the geometric error made if the domain is a Riemannian manifold ratherthan a polyhedral domain. The approach involves the development of variationalcrimes analysis in Hilbert complexes, and then using the abstract framework todevelop analogues of the Strang Lemmas for the Finite Element Exterior Calculus(FEEC). We indicate how this variational crimes framework in FEEC recovers theclassical a priori surface finite element estimates of Dziuk and Demlow, and allowsfor substantial generalizations, including hypersurfaces of arbitrary spatial dimen-sion, the Hodge Laplacean, nonlinear problems, as well as semilinear parabolic andhyperbolic problems. This is joint work with a number of colleagues over the lastthree years.
34
Mini Symposium M1Finite Element Packages with Domain Decom-position SolversSchedule Author Index Session Index
Organizers: Frederic Hecht, Frederic Nataf, Christophe Prud’homme
AbstractMost linear solvers fall into one of these four categories: direct solvers, incompletefactorizations, multigrid methods or domain decomposition methods. Domain de-composition methods are naturally parallel. They are unique in the sense that theycan be thought of in terms of partial differential equations, often in their variationalforms. For this reason they are natural to implement and use in finite element pack-ages such as Freefem++ or Feel++ (formerly known as Life). We present recentdevelopments in this direction that enable these packages to address large scaleproblems on clusters or HPC platforms. We explain how to implement domaindecomposition methods in these frameworks and give numerical examples.
35
M1 Schedule
Finite Element Packages with Domain DecompositionSolvers
Schedule Author Index Session Index
Date: Friday, June 29Time: 10:30-12:15Location: MarkovChairman: Frederic Hecht, Frederic Nataf, Christophe Prud’homme
10:35-11:00 : Frederic HechtSome Ways to Implement Domain Decomposition Methods in Freefem++Abstract
11:00-11:25 : Pierre JolivetMultilevel Spectral Coarse Space Methods in Freefem++Abstract
11:25-11:50 : Christophe Prud’hommeDomain Decomposition Methods in Feel++Abstract
11:50-12:15 : Abdoulaye SamakeSubstructuring Preconditioners for the Mortar Method in Feel++Abstract
36
Some Ways to Implement Domain Decomposition Methods inFreefem++
Session Schedule Author Index Session Index M1
Frederic HechtUPMC Univ Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions, F-75005, Paris,[email protected]
AbstractWe present how to solve Poisson equation with Schwarz method, in FreeFem++software. The Poisson problem on domain Ω with boundary Γ in L2(Ω) :is
−∆u = f, in Ω, and u = g on Γ,
where f and g are two given functions of L2(Ω) and of H12 (Γ), Let introduce (πi)i∈I
a regular positive partition of the unity of Ω in Np = #I < +∞ , and denote Ωi thesub domain which is the support of πi function and also denote Γi the boundary ofΩi.The parallel Schwarz method with overlapping is Let ` = 0 the iterator and aninitial guest u0 respecting the boundary condition (i.e. u0
|Γ = g).
∀i ∈ I −∆u`i = f, in Ωi, and u`i = u` on Γi (1)
u`+1 =∑i∈I
πiu`i (2)
After discretization with the Lagrange finite element method, with a compatiblemesh Thi of Ωi, i. e., the exist a global mesh Th such that Thi is include in Th.The problem is find
∀i ∈ I, ∀vhi ∈ V0hi
∫Ωi
∇u`hi.∇vhi =
∫Ωi
fvhi, and u`hi = u`h on Γi (3)
u`+1h =
∑i∈I
πiu`hi (4)
We show how to the solve the problem (3 − 4) by four methods: a basic Schwarzalgorithm (for teaching), a first acceleration with a GMRES algorithm to computethe solution of the engine problem : find u` such that it’s equal to the next iterateu`+1 = u` , and two classical Restricted Additive Schwarz methods with and withouta coarse grid preconditioned.Finally, we try these algorithms on Elasticity problem and Stokes problem and showsome numerical result of this problem.
37
Multilevel Spectral Coarse Space Methods in FreeFem++
Session Schedule Author Index Session Index M1
Pierre JolivetLaboratoire Jacques-Louis Lions, ParisLaboratoire Jean Kuntzmann, [email protected]
Victorita DoleanLaboratoire Jean-Alexandre Dieudonne, [email protected]
Frederic HechtLaboratoire Jacques-Louis Lions, [email protected]
Frederic NatafLaboratoire Jacques-Louis Lions, [email protected]
Nicole SpillaneLaboratoire Jacques-Louis Lions, [email protected]
AbstractCoarse space correction is essential to achieve algorithmic scalability in domaindecomposition methods. Our goal here is to build robust coarse spaces for Schwarz-type preconditioners for elliptic problems with highly heterogeneous coefficientswhen the discontinuities are not just across but also along subdomain interfaces,using local spectral information based on an analysis of the underlying partial dif-ferential equations.This construction is then implemented within the C++ domain specific languageFreeFem++, and the numerical efficiency of our method is assessed on large-scalecomputer architectures.
38
Domain Decomposition Methods in Feel++
Session Schedule Author Index Session Index M1
Christophe Prud’hommeUniversite de Grenoble 1 / CNRS, Laboratoire Jean Kuntzmann / UMR 5224,Grenoble, F-38041, [email protected]
Abdoulaye SamakeUniversite de Grenoble 1 / CNRS, Laboratoire Jean Kuntzmann / UMR 5224,Grenoble, F-38041, [email protected]
Vincent ChabannesUniversite de Grenoble 1 / CNRS, Laboratoire Jean Kuntzmann / UMR 5224,Grenoble, F-38041, [email protected]
Christophe PicardUniversite de Grenoble 1 / CNRS, Laboratoire Jean Kuntzmann / UMR 5224,Grenoble, F-38041, [email protected]
AbstractWe present our advances in domain decomposition methods including shwartz, mor-tar and three fields methods by showing that the embedding language in C++ calledFeel++ goes little in the way of expressivity and closeness to the mathematicallanguage . First we will focus on the overlapping and nonoverlapping schwartzmethods with and without relaxation and the Aitken procedure to compute therelaxation parameter. And then we turn to the nonconforming formulations of do-main decomposition methods: the mortar method where the main idea is to enforcethe weak continuity between the solutions on each subdomain by introducing a La-grange multiplier corresponding to this connection constraint and then the threefields method in which the disadvantage of introducing the third unknown field iscompensated for by the fact that all subdomains are treated exactly in the sameway, which results in an easier implementation and possibly, when considering theparallelization of the method, in an easier balancing of the load between processors.The numerical tests will support the above domain decomposition methods usingthe Feel++ framework. The numerical tests are also in parallel and we rely onPETSc.
39
Substructuring Preconditioners for the Mortar Method inFeel++
Session Schedule Author Index Session Index M1
Abdoulaye SamakeUniversite de Grenoble 1 / CNRS, Lab. Jean Kuntzmann / UMR 5224, Grenoble,F-38041, [email protected]
Sivia BertoluzzaIstituto di Matematica Applicata e Tecnologie Informatiche del CNR “Enrico Ma-genes”[email protected]
Micol PennacchioIstituto di Matematica Applicata e Tecnologie Informatiche del CNR “Enrico Ma-genes”[email protected]
Christophe Prud’hommeUniversite de Grenoble 1 / CNRS, Lab. Jean Kuntzmann / UMR 5224, Grenoble,F-38041, [email protected]
AbstractWe deal with the efficient solution of the linear system arising from the discretiza-tion by the mortar method, a nonconforming version of the domain decompositionmethods, in two and three dimensions. This kind of preconditioners has alreadybeen applied to the mortar methods in two dimensions or the case of order onefinite elements and in an abstract framework including high order finite elementsand wavelets. We focus on the simple model problem: find u : Ω → R, with Ωbounded polyhedral domain of R3, verifying −∇ · a∇u = f in Ω, u = 0 on ∂Ω.We split the discrete solution uh ∈
∏X`h as the sum of three suitable constructed
contributions uh = u0h + uFh + uWh , with u0
h ∈∏X`h ∩ H1
0 (Ω`) corresponding tonodes interior to the subdomains, uFh corresponding to nodes interior to the facesof the subdomains, and uWh corresponding to nodes on the wirebasket. We pro-
pose a block diagonal preconditioner A−1 for the mortar method. More preciselyit is defined as the matrix corresponding to a bilinear form defined as a(uh, vh) :=a0(u0
h, v0h)+ aF (uFh , v
Fh )+ aW (uWh , v
Wh ). The two bilinear forms a0 and aF are them-
selves block diagonal, the blocks corresponding respectively to the subdomains andto the faces composing the interface. We shall show that the overall system en-
joys cond(A−1A) ≤ C(
1 + max` log H`
h`
)4
. where ` is a subdomain index. We willpresent an analysis of this preconditioner as well as numerical results obtained withthe Feel++ framework.
40
Mini Symposium M2Domain Decomposition for Porous Media Flowand TransportSchedule Author Index Session Index
Organizers: Caroline Japhet and Michel Kern
AbstractPorous media flow and transport have many applications such as far field simula-tions of underground nuclear waste disposal, geological storage of CO2, or reservoirengineering... A salient feature of subsurface flow and transport processes is theheterogeneity of the medium with physical properties ranging over several ordersof magnitude. Other challenges presented by these models involve widely differingspace-time scales. Accurately resolving these features requires fines meshes, andthus the solution of large systems. Domain decomposition methods are a very im-portant part of a solution procedure. The aim of this minisymposium is to bringtogether scientists working in this field to report about recent developments. Workpresented will range from space-time methods, to model coupling for multiphaseflow, algebraic solution procedures, numerical zoom preconditioners, parallel simu-lators and non-linear extensions of the framework.
Part 1
Part 2
41
M2P1 Schedule
Domain Decomposition for Porous Media Flow and Trans-port
Schedule Author Index Session Index
Date: Wednesday, June 27Time: 10:30-12:15Location: AmphiChairman: Caroline Japhet and Michel Kern
10:35-11:00 : Oliver SanderDiscretizations for the Richards Equation Based on KirchhoffTransformationAbstract
11:00-11:25 : Thi Thao Phuong HoangSpace-Time Domain Decomposition For Mixed Formulations of TransportProblems In Porous MediaAbstract
11:25-11:50 : Frederic NatafAlgebraic Domain Decomposition Methods for Highly HeterogeneousProblemsAbstract
11:50-12:15 : Zhangxin ChenGPU-based Parallel Reservoir SimulatorsAbstract
M2 Abstract
Part 2
42
Discretizations for the Richards Equation Based on KirchhoffTransformation
Session Schedule Author Index Session Index M2
Oliver SanderFreie Universitat [email protected]
Heiko BerningerUniversite de [email protected]
Ralf KornhuberFreie Universitat [email protected]
AbstractWe consider a discretization of the Richards equation constructed by first apply-ing the Kirchhoff transformation and then discretizing the transformed equationusing first-order finite elements. If the permeability and saturation functions areindependent of space only with respect to a partition of the domain, we transformseparately on each subdomain and combine the transformed subproblems by non-linear transmission conditions. We give various characterizations of the resultingdiscretizations, and demonstrate optimal a priori error bounds. The discretizationsare solver-friendly in the sense that each subdomain problem becomes equivalentto a convex minimization problem, which can be solved efficiently using monotonemultigrid methods. For the overall problem we use different substructuring meth-ods, and show that they are robust with respect to large variations of the soilparameters.
43
Space Time Domain Decomposition Methods For MixedFormulations of Transport Problems In Porous Media
Session Schedule Author Index Session Index M2
Thi Thao Phuong HoangProject Pomdapi, INRIA Rocquencourt, FrancePhuong.Hoang Thi [email protected]
Jerome JaffreProject Pomdapi, INRIA Rocquencourt, [email protected]
Caroline JaphetLAGA, University Paris 13 and Project Pomdapi, INRIA, [email protected]
Michel KernProject Pomdapi, INRIA Rocquencourt, [email protected]
Jean RobertsProject Pomdapi, INRIA Rocquencourt, [email protected]
AbstractThe far field simulation of underground nuclear waste disposal site requires a highcomputational cost due to the widely varying properties of different materials, thedifferent length and time scales, and the high accuracy requirements. Nonoverlap-ping domain decomposition methods allow local adaptation in both space and timeand result in parallel algorithms. We have extended the optimized Schwarz wave-form relaxation (OSWR) method, successfully used for finite elements and finitevolumes, to the case of mixed finite elements with their local mass-conservationproperty. Another choice is the substructuring method, which has been shown tobe efficient for steady state problems with strong heterogeneities. We study a time-dependent Schur complement method, which is the algebraic counterpart of thediscrete Steklov Poincare operator, and introduce the Neumann preconditioner aswell as weight matrices (following work of De Roeck Le Tallec) designed to make theconvergence speed independent of the heterogeneities. Both methods enable the useof local time steps when the subdomains have highly different physical properties.Their performance is illustrated on test cases suggested by nuclear waste disposalproblems. This work is supported by ANDRA, the French Agency for NuclearWaste Management.
44
Algebraic Domain Decomposition Methods for HighlyHeterogeneous Problems
Session Schedule Author Index Session Index M2
Frederic NatafLaboratoire J.L. Lions, UPMC and CNRS, Paris, [email protected]
Pascal HaveIFP, [email protected]
Roland MassonUniversite de Nice, [email protected]
Mikolaj SzydlarskiIFP, [email protected]
Tao ZhaoLaboratoire J.L. Lions, UPMC and CNRS, Paris, [email protected]
AbstractWe consider the solving of linear systems arising from porous media flow simulationswith high heterogeneities. Using a Newton algorithm to handle the non-linearityleads to the solving of a sequence of linear systems with different but similar ma-trices and right hand sides. The parallel solver is a Schwarz domain decompositionmethod. The unknowns are partitioned with a criterion based on the entries of theinput matrix. This leads to substantial gains compared to a partition based onlyon the adjacency graph of the matrix. From the information generated during thesolving of the first linear system, it is possible to build a coarse space for a two-leveldomain decomposition algorithm that leads to an acceleration of the convergence ofthe subsequent linear systems. We compare two coarse spaces: a classical approachand a new one adapted to parallel implementation.
45
GPU-based Parallel Reservoir Simulators
Session Schedule Author Index Session Index M2
Zhangxin ChenUniversity of [email protected]
Hui LiuUniversity of [email protected]
Song YuUniversity of [email protected]
Ben HsiehUniversity of [email protected]
Lei ShaoUniversity of [email protected]
AbstractLarge-scale reservoir simulation demands significant computational time so improv-ing its computational efficiency becomes crucial. Graphics Processing Unit (GPU), ahigh-profile parallel processor with hundreds of microprocessors, offers great poten-tial in parallel reservoir simulation because of its efficient power utilization and highcomputational efficiency. In addition, its cost is relatively low, making large-scaleparallel reservoir simulation possible for most of desktop users. In this presentationseveral GPU-based parallel linear solvers and preconditoners will be discussed. Theyinclude the GMRES, BiCGSTAB and ORTHOMIN solvers and the incomplete LU(ILU) factorization, domain decomposition and algebraic multigrid preconditioners.These solvers and preconditoners have been coupled with an in-house black-oil sim-ulator to speedup reservoir simulation. In the numerical experiments performed,the SPE 10 problem, a 3D heterogeneous benchmark model with over one milliongrid blocks, is selected to test the speedup of the resulting black-oil simulator. Onthe state-of-the-art CPU and GPU platforms, the new GPU implementation canachieve a speedup of over eight times in solving linear systems arising from thisSPE 10 problem compared with the CPU implementation.
46
M2P2 Schedule
Domain Decomposition for Porous Media Flow and Trans-port
Schedule Author Index Session Index
Date: Wednesday, June 27Time: 16:00-17:45Location: AmphiChairman: Caroline Japhet and Michel Kern
16:05-16:30 : Bernd FlemischModel Coupling for Multiphase Flow in Porous MediaAbstract
16:30-16:55 : Paul-Marie BertheSpace-Time Domain Decomposition with Finite volumes for Porous MediaApplicationsAbstract
16:55-17:20 : Jean-Baptiste Apoung KamgaA Numerical Zoom Preconditioner for Discontinuous Galerkin DomainDecomposition Approximation of Darcy FlowAbstract
17:20-17:45 : Anthony MichelTime Space Domain Decomposition for Reactive Transport in Porous Media.Application to CO2 Geological StorageAbstract
M2 Abstract
Part 1
47
Model Coupling for Multiphase Flow in Porous Media
Session Schedule Author Index Session Index M2
Bernd FlemischIWS-LH2, University of [email protected]
Rainer HelmigIWS-LH2, University of [email protected]
AbstractNumerical models for flow and transport in porous media are valid for a particularset of processes, scales, levels of simplification and abstraction, grids etc. Thecoupling of two or more specialised models is a method of increasing the overall rangeof validity while keeping the computational costs relatively low. Several couplingconcepts are reviewed in this talk with a focus on the authors work in this field.The concepts are divided into temporal and spatial coupling concepts, of which thelatter is subdivided into multi-process, multi-scale, multi-dimensional, and multi-compartment coupling strategies. Examples of applications for which these conceptscan be relevant include groundwater protection and remediation, carbon dioxidestorage, nuclear-waste disposal, soil dry-out and evaporation processes as well asfuel cells and technical filters. In particular, we focus on the coupling of single-phasecompositional non-isothermal free flow and two-phase porous media flow.
48
Space-Time Domain Decomposition with Finite Volumes forPorous Media Applications
Session Schedule Author Index Session Index M2
Paul-Marie BertheCEA, DEN, DM2S-STMF-LMSF and University Paris 13, [email protected]
Pascal OmnesCEA, DEN, [email protected]
Caroline JaphetLAGA, University Paris 13 and Project Pomdapi, [email protected]
AbstractIn the context of porous media applications such as nuclear waste repositories, weconsider the solving of time dependent advection-diffusion problems with possiblyhighly discontinuous coefficients, modelling radionucleides flow and transport in theunderground. Because of the widely varying properties of the different materials,one need to use different time step and mesh size in different regions of the compu-tational domain. The Optimized Schwarz Waveform Relaxation (OSWR) method,which has been developed over the last decade, is one possible strategy since itallows to use different space-time discretizations in subdomains, possibly noncon-forming and needs a very small number of iterations to converge. This method hasbeen analyzed with discontinous Galerkin method for the time discretization, sothat rigorous analysis can be made for any degree of accuracy, and time steps canbe adaptively controlled by a posteriori error analysis. On the other hand, using aDiscrete Duality Finite Volume (DDFV) method allows to use any type of meshes inspace, including highly non conforming meshes. Thanks to the discrete variationalformulation of the scheme, local refinement of the meshes can be done with efficienta posteriori error estimators.We design and study an extension of the DDFV scheme to time dependent advection-diffusion problems in the context of optimized Schwarz waveform relaxation, with atime discontinous Galerkin method. We propose a new discretization of the fluxes,which permit to prove the well-posedness of the nonconforming in time domain de-composition method and the convergence of the iterative solver to the global onedomain scheme.
49
A Numerical Zoom Preconditioner for Discontinuous GalerkinDomain Decomposition Approximation of Darcy Flow
Session Schedule Author Index Session Index M2
Jean-Baptiste Apoung KamgaUniv Paris-Sud, Laboratoire de Mathematiques, UMR 8628, Orsay [email protected]
AbstractMixed discontinous Galerkin methods are well suited for the approximation of DarcyFlow when accurate approximation of the velocity is required. These methods areunfortunately too much memory consuming and can thus benefit from parallel pro-gramming in a distributed memory environment. But because of their modal ap-proximation nature, their substructuration in the framework of non overlappingdomain decomposition method is not straightforwards. In the present, by introduc-ing the notion of interface skeleton, a substructure procedure in designed and theacceleration of the solution of the substructured problem is preformed with the helpof the numerical zoom techniques. Two and three dimensional numerical tests arefurnished for illustration.
50
Time Space Domain Decomposition for Reactive Transport inPorous Media. Application to CO2 Geological Storage
Session Schedule Author Index Session Index M2
Anthony MichelIFP Energies [email protected]
Florian HaeberleinIFP Energies nouvelles and LAGA Universite Paris [email protected]
Laurence HalpernLAGA Universite Paris [email protected]
AbstractNumerical modeling is the main way to reduce uncertainties about the long termevolution of CO2 storage in the underground. When super-critical CO2 is injectedin an aquifer, it dissolves in brine, modifying its chemical properties. The result-ing aqueous solution may react with the host rocks in the reservoir, the cementsaround the wells or the shale into the cap-rock. In order to predict the locationand amount of rock modifications, we have developed coupled reactive transportand multiphase flow models. Up to now, these simulations are always restricted tocoarse grids because they are too expensive in term of CPU-Time. However, weknow that the main numerical difficulties are usually localized in space and time.As a consequence, these difficulties may be strongly reduced if we could isolate thereactive zones and solve them separately. SWR time space domain decompositiontechniques have already proved their efficiency in solving such a problem for scalarlinear convection-diffusion equations and recently for semi-linear scalar problems.However, there remains to prove it for system of nonlinear equations like realisticmulti-species reactive transport problems.With the support of the ANR-SHPCO2 project, we have implemented a SWRdomain decomposition framework and a global reactive transport module in theCEA-IFPEN parallel platform Arcane. By allowing the domain decomposition inspace to be dynamic and the time step to be adapted by sub-domain, we have beenable to track the reactive zones. Even if these results are already impressive, wehaven’t been able to measure the effective gain of performance of this strategy. Asa consequence, a new performance study is necessary.In this paper we will present the last performance results obtained for a few studytest cases and discuss their sensitivity to the mesh, the size of the overlap and theoptimized parameters.
51
52
Mini Symposium M3Finite Elements for First-Order System Formu-lations of Interface ProblemsSchedule Author Index Session Index
Organizers: Pavel Bochev and Gerhard Starke
AbstractLeast squares finite element methods for first-order system formulations of modelsin fluid and solid mechanics have become increasingly popular in recent years. Suchmethods allow simultaneous approximation of all process variables by finite elementspaces that are not subject to joint stability (inf-sup) conditions. Norm-equivalenceof least-squares functionals leads to symmetric and positive definite linear systemsand optimal error estimates with respect to suitable error norms. These advantageslead to simpler formulations of the coupling conditions arising in connection tointerface problems. The talks in this minisymposium are concerned with differentaspects of the treatment of coupling conditions for interface problems in the contextof first-order system formulations. This includes the treatment of moving interfacesmodeled by phase-field or level-set methods. An important issue for numericalapproximations of flow models is the accuracy of mass conservation. This will alsobe a central topic in these contributions. The session will also address iterativemethods based on domain decomposition ideas for such interface models.
53
M3 Schedule
Finite Elements for First-Order System Formulations ofInterface Problems
Schedule Author Index Session Index
Date: Thursday, June 28Time: 10:30-12:15Location: TuringChairman: Pavel Bochev and Gerhard Starke
10:35-11:00 : James AdlerConstrained First-Order System Least Squares for Improved MassConservationAbstract
11:00-11:25 : Pavel BochevLeast-Squares Methods for Mesh-TyingAbstract
11:25-11:50 : Fleurianne BertrandLeast Squares Methods with Interface Approximation for Two Phase StokesFlowAbstract
11:50-12:15 : Steffen MunzenmaierLeast Squares Finite Element Methods for Coupled Generalized NewtonianStokes-Darcy FlowAbstract
54
Constrained First-Order System Least Squares for ImprovedMass Conservation
Session Schedule Author Index Session Index M3
James AdlerTufts [email protected]
Panayot S. VassilevskiLawrence Livermore National [email protected]
AbstractIn complex fluid flow simulations, there is a tradeoff between obtaining solutionsthat are accurate with a reasonable amount of computational work and satisfyingcertain conservation laws exactly. For instance, in incompressible fluid flow, conser-vation of mass takes the form of making sure the fluid velocities are divergence-free.In magnetohydrodynamics, one must satisfy conservation of mass as well as thesolenoidal constraint that the magnetic field is divergence-free (i.e. there are nomagnetic monopoles). Many methods have been applied to such systems, somebeing conservative at the cost of accuracy of the momentum equations and othersat the cost of efficiency in the solver. First-order system least-squares approacheshave also been applied and yield efficient methods for approximating solutions tocoupled fluid mechanics problems. However, without proper care, the auxiliaryconservation equations may not be solved to a sufficient accuracy. In this talk,we propose a constrained least-squares approach, where we augment the first-ordersystem and minimize the least-squares functional subject to some constraint. Here,we only look at a simple diffusion equation, but present the main ideas, includingwhat types of finite-element spaces to use and the solution algorithm. A domaindecomposition or multilevel approach is employed to solve the constrained problemon local subdomains and coarse grids and used to update the unconstrained solutionas needed. Thus, we approximate the solution accurately and efficiently using theleast-squares method, while still conserving the appropriate quantity.
55
Least-Squares Methods for Mesh-Tying
Session Schedule Author Index Session Index M3
Pavel BochevNumerical Analysis and Applications, MS1320, Sandia National Laboratories1, P.O.Box 5800, Albuquerque, New Mexico 87185, [email protected]
Oxana GubaNumerical Analysis and Applications, MS1320, Sandia National Laboratories, P.O.Box 5800, Albuquerque, New Mexico 87185, [email protected]
AbstractMesh tying refers to the finite element solution of Partial Differential Equations(PDEs) on a union of independently meshed subdomains. This task arises in com-putational modeling of scientific and engineering problems posed on domains withcomplex geometries. Efficient grid generation for such domains often requires sep-arate meshing of their parts. If the interfaces between the parts are curved, theirindependent meshing generally leads to adjoining surfaces that do not coincide spa-tially. A minimal requirement for any mesh-tying method is a consistency conditioncalled patch test. A method passes a patch test of order k if it can recover solutionsof the governing equations that are global polynomials of degree k. Mesh-tying for-mulations based on standard Galerkin methods experience difficulties passing suchtests because the presence of gaps and overlaps between the domain parts causesphysical energy to be undercounted or overcounted. As a result, most state of theart methods only pass patch tests of order 1, i.e., they preserve at most globallylinear solutions. In this talk we present an alternative approach which utilizes least-squares variational principles. A least-squares functional is a sum of the residualsof the PDEs measured in Sobolev space norms. As a result, such a functional al-ways vanishes at the exact solution. By exploiting this property, we formulate aleast-squares method for mesh-tying, which automatically passes a patch test ofthe same order as the finite element space employed in its definition. Specifically,because in mesh-tying applications the non-coincident interfaces are close, small in-terface perturbations eliminate the voids and create overlapping domains. Then, bymeasuring residual energy and not physical energy, a least-squares functional maymeasure energy redundantly in those subdomain intersections. Numerical resultsdemonstrate the potential of this idea.
1Sandia National Laboratories is a multi-program laboratory managed and operated by SandiaCorporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the U.S. Departmentof Energys National Nuclear Security Administration under contract DE-AC04-94AL85000.
56
Least Squares Methods with Interface Approximation for TwoPhase Stokes Flow
Session Schedule Author Index Session Index M3
Fleurianne BertrandLeibniz Universitat [email protected]
Gerhard StarkeLeibniz Universitat [email protected]
AbstractWe consider the coupled problem with Stokes flow in two subdomains sepa-
rated by an interface. At the interface, continuity of the velocity and the momentumbalance condition for the stress tensor need to be imposed. The interface is char-acterized by a level set function which satisfies an appropriate transport equationand the problem can be written as a domain decomposition problem.
In this talk we first present how the stationary Stokes problem can be writtenas a first order system. For numerical results a combination of H(div)-conformingRaviart-Thomas and standard H1-conforming elements is used.
After that we analyze the effect of approximated flux boundary conditionson Raviart-Thomas finite elements in order to get the effect of the approximatedinterface on the momentum balance condition. In particular, we present an estimatefor the normal flux on interpolated boundaries.
57
Least Squares Finite Element Methods for Coupled GeneralizedNewtonian Stokes-Darcy Flow
Session Schedule Author Index Session Index M3
Steffen MunzenmaierInstitute of Applied Mathematics, Leibniz University [email protected]
Gerhard StarkeInstitute of Applied Mathematics, Leibniz University [email protected]
AbstractIn this talk we consider a coupled Stokes-Darcy flow problem. The domain isdecomposed into a fluid and a porous media region. In the fluid region the problemis given by the Stokes equations for generalized Newtonian flow whereas in theporous media a Darcy equation for generalized Newtonian flow has to be solved.Both flows are treated as a first order system in a pseudostress-velocity formulationfor the Stokes problem and a volumetric flux-hydraulic potential formulation for theDarcy problem. The coupling along an interface is done by using the well knownBeavers-Joseph-Saffman interface condition. A least squares finite element methodis used for the numerical approximation of the solution. It will be shown thatthe least squares functional corresponding to the nonlinear first order system is anefficient and reliable error estimator which allows for adaptive refinement.
58
Mini Symposium M4On the Origins of Domain Decomposition Methods
Schedule Author Index Session Index
Organizers: Martin J. Gander
AbstractDomain decomposition methods have been developed in various contexts, and withvery different goals in mind. This minisymposium has the purpose to show indetail where some of the most successfull domain decomposition methods comefrom, and why they have been invented. It will trace the history of the alternatingSchwarz method, which fixed an important gap in Riemann’s proof of the famousRiemann mapping theorem, the development of substructuring in the engineeringcommunity at Boeing by Przemieniecki around the same time of the re-invention ofthe finite element method, the formulation of the FETI (Finite element Tearing andInterconnect) method in Paris and its evolution, and the revolution brought by theinvention of the additive Schwarz method and the abstract Schwarz framework.
59
M4 Schedule
On the Origins of Domain Decomposition Methods
Schedule Author Index Session Index
Date: Friday, June 29Time: 10:30-12:15Location: I50Chairman: Martin J. Gander
10:35-11:00 : Martin J. GanderOn the Origins of the Alternating Schwarz MethodAbstract
11:00-11:25 : Xuemin TuOrigin of Iterative Substructuring MethodsAbstract
11:25-11:50 : Francois-Xavier RouxFETI: Finite Element Tearing and InterconnectingAbstract
11:50-12:15 : Olof WidlundEarly Work on Two-Level Schwarz Algorithms and where it had led usAbstract
60
On the Origins of the Alternating Schwarz Method
Session Schedule Author Index Session Index M4
Martin J. GanderUniversity of [email protected]
AbstractMany people know that Hermann Amandus Schwarz invented his alternating methodin order to show that solutions to Laplace’s equation exist on domains with quitecomplicated shapes; but why was this so important in 1869? We answer this ques-tion by going back to Riemann’s PhD thesis from 1851, where on the last pages,one can find the famous Riemann Mapping Theorem. In the proof, which is con-structive, there is an important gap, and when Riemann was challenged, he simplyreplied that to close the gap, it suffices to use the Dirichlet principle, which Dirich-let was teaching in his lectures to undergraduate students during the same time.Unfortunately, Weierstrass quickly found a counterexample, in which the Dirichletprinciple led to an incorrect conclusion. While Riemann replied to Weierstrass thathis theorem nevertheless is true, an international competition was launched to closethe gap in the proof of the Riemann mapping theorem. This theorem can be con-sidered to be the foundation of Riemann’s theory of analytic functions, and it wasSchwarz who managed to close this gap with his alternating method.
61
Origin of Iterative Substructuring Methods
Session Schedule Author Index Session Index M4
Xuemin TuUniversity of [email protected]
Martin J. GanderUniversity of [email protected]
AbstractIterative substructuring methods (Schur complement methods) are non-overlappingdomain decomposition methods. The name is borrowed from the structural engi-neering community back to 1970s. These methods are preconditioned iterativemethods for the subdomain boundary values (primal methods) or the normal deriva-tives (dual methods). The active development of these methods was started fromthe early 1990s. The most popular methods among this class are BDDC and FETI–DP algorithms which originate from the balancing NN and one-level FETI methods.In this talk, we will trace these methods back to 1963 by Przemieniecki and give areview of the development.
62
FETI: Finite Element Tearing and Interconnecting
Session Schedule Author Index Session Index M4
Francois-Xavier [email protected]
AbstractIn this paper we recall the origin of the FETI method, initially derived from mixedfinite element methods with discontinuous solutions and Lagrange multipliers forenforcing the continuity condition. The fundamental ingredients of the originalFETI method, the use of discrete, eventually redundant Lagrange multipliers, theissue with ill-posed local Neumann problems that leads to the natural coarse gridprojector of FETI, are presented. The two-level FETI method ideas with enrichmentof the coarse space is introduced with its main drawback due to the fact that thecoarse operator is not sparse. Then the best known extensions of the method thatlead to a sparse coarse preconditioner, the FETI-DP and the FETI-2LM methodsare presented.
63
Early Work on Two-Level Schwarz Algorithms and where it hadled us
Session Schedule Author Index Session Index M4
Olof B. WidlundCourant Institute, New York, New [email protected]
AbstractAround the time of DD1, it was realized that the classical Schwarz algorithm couldbe improved by adding an additional coarse level solver. In addition, an alternativeadditive variant was introduced to simplify the parallelization of these iterativemethods; the additive variant had in fact been introduced earlier in Novosibirsk byMatsokin and Nepomnayaschikh. The additive algorithms inspired the developmentof an abstract Schwarz frame work, which has been important to the developmentof many domain decomposition algorithms. The original algorithms required thatthe fine mesh is a refinement of a coarse conventional finite element model. Progressin removing this assumption will be discussed including the introduction of coarsemethods based on quite irregular coarse basis elements and the successful use ofthese algorithms for almost incompressible elasticity. The family of abstract Schwarzmethods has also been extended and several hybrid methods of this kind will bepresented.
64
Mini Symposium M5Exotic Coarse Spaces for Domain Decomposi-tion MethodsSchedule Author Index Session Index
Organizers: Martin J. Gander, Laurence Halpern, Kevin Santugini
AbstractIn the absence of a coarse grid, domain decomposition methods can only exchangeinformation between adjacent subdomains, and hence such methods can never con-verge in less iterations than the diameter of the connectivity graph between sub-domains. Domain decomposition methods without a coarse grid can therefore notbe scalable. By adding a coarse grid, it becomes possible to transfer informationglobally at each iteration, and one can then hope to obtain a scalable method, i.e.a method where the number of iteration does not (or only weakly) depend on thenumber of subdomains. But how should such a coarse grid be constructed ? Is itreally necessary to base the long range information exchange on a grid, or is it alsopossible to construct a mechanism of long range interaction by other coarse spacecomponents?This minisymposium gives you an overview of developments in this direction overthe last few years. Coupling coarse grids with elaborate domain decompositionmethods has proved to be difficult, and many other techniques can be found in theliterature to add coarse space components to domain decomposition algorithms.Some of these attempts are based on algebraic considerations like deflation, somework directly on discrete subsets of the unknowns, like wirebasket techniques, andagain others try to reason at the continuous level. Designing efficient domain de-composition algorithms with a coarse space requires great care and insight, and isa major area of current research in domain decomposition methods.
65
M5 Schedule
Exotic Coarse Spaces for Domain Decomposition Meth-ods
Schedule Author Index Session Index
Date: Monday, June 25Time: 10:30-12:15Location: I50Chairman: Martin J. Gander, Laurence Halpern, Kevin Santugini
10:35-11:00 : Martin J. GanderA new Coarse Grid Correction for RASAbstract
11:00-11:25 : Clark DohrmannLower Dimension Coarse Spaces for Overlapping Schwarz AlgorithmsAbstract
11:25-11:50 : Jorg WillemsSpectral Coarse Space Construction in Robust Multilevel MethodsAbstract
11:50-12:15 : Kevin SantuginiDiscontinuous Coarse Space Corrections (DCS) for Optimized SchwarzMethodsAbstract
66
A new Coarse Grid Correction for RAS
Session Schedule Author Index Session Index M5
Martin J. GanderUniversity of [email protected]
Laurence HalpernUniversity Paris [email protected]
Kevin SantuginiInstitut Polytechnique de [email protected]
AbstractRestricted Additive Schwarz (RAS) has become one of the most popular Schwarzmethod, especially for non-symmetric problems. In contrast to Additive Schwarz,there is however no comprehensive convergence theory available for RAS: for theone level variant, there is a proof of equivalence to the classical parallel Schwarzmethod introduced by Lions. We are interested in the precise interplay between theRAS iterates and the coarse grid correction. We show that for a classical coarsegrid correction with one (or a few) nodes within each subdomain, the interactionwith the RAS iterates can lead to oscillations which are very unfavorable for theconvergence of the method. Placing the coarse grid correction nodes into the overlapremoves these oscillations and leads to an overall much faster two level method. Forone dimensional problems, one can even obtain a method which converges in oneiteration, i.e. one parallel subdomain solve, and one coarse grid correction, providedthe coarse grid shape functions have certain properties.
67
Lower Dimension Coarse Spaces for Overlapping SchwarzAlgorithms
Session Schedule Author Index Session Index M5
Clark DohrmannSandia National Laboratories, Albuquerque, New Mexico, [email protected]
Olof B. WidlundCourant Institute, New York, New York, [email protected]
AbstractWe present some new coarse spaces for overlapping Schwarz algorithms of lowerdimension than earlier ones. The first one is a vertex-based coarse space for scalarproblems in the plane which is significantly smaller than a previous one based onboth subdomain vertices and edges. In addition, the condition number estimateassociated with the vertex-based coarse space is improved by a factor of log(H/h)in comparison to the estimate for the richer coarse space. Lower dimension coarsespaces and some theoretical results are also presented for three dimensional prob-lems. In comparison to iterative substructuring methods such as FETI-DP orBDDC, a vertex only coarse space does not suffer from a linear factor of H/hin condition number estimates. Numerical examples are presented to confirm thetheory and to demonstrate the utility of the coarse spaces.
68
Spectral Coarse Space Construction in Robust MultilevelMethods
Session Schedule Author Index Session Index M5
Joerg WillemsRadon Institute for Computational and Applied Mathematics (RICAM)[email protected]
AbstractIn the design of robust two-level domain decomposition methods the choice of thecoarse spaces is crucial. Here the term “robust” refers to convergence rates whichare independent of problem and mesh parameters. Important instances of suchproblem parameters are in particular (highly) varying coefficients. For symmetricpositive definite (SPD) systems the objective of achieving robustness with respectto mesh parameters has been successfully addressed by e.g. two-level alternatingSchwarz methods employing standard coarse spaces. For arbitrarily general config-urations robustness with respect to (large) coefficient variations has proved to be amore complicated goal.
For the stationary heat equation Efendiev and Galvis introduced a coarsespace construction based on local generalized eigenvalue problems. Using only thoseeigenmodes in the coarse space construction corresponding to eigenvalues below apredefined threshold resulted in an overlapping additive Schwarz method yieldinga condition number independent of variations in the coefficients. The approach ofusing local spectral problems for constructing coarse spaces was then further gen-eralized to abstract SPD operators in a paper to which the author contributed.
In the present contribution we consider the generalization of the two-levelapproach to a multilevel method. This extension, which is carried out in the frame-work of (nonlinear) algebraic multilevel iterations (AMLI), is mainly motivated bythe high numerical cost for solving generalized eigenvalue problems, which limits theapplicability of the two-level method, since either the local generalized eigenvalueproblems or the global coarse problem become too big. Our approach is analyzedin a rather general setting, which is shown to be applicable to the stationary heatequation, the equations of linear elasticity, and equations arising in the solution ofMaxwell’s equations.
Analogous to the two-level method the crucial ingredient in our approach isthe construction of a hierarchy of spaces VL ⊂ VL−1 ⊂ . . . ⊂ Vl ⊂ Vl−1 ⊂ . . . ⊂ V0
based on generalized eigenvalue problems. Our analysis essentially relies on an in-exact stable decomposition property for two consecutive spaces Vl+1 ⊂ Vl with aconstant that is independent of problem and mesh parameters. Based on this we de-fine a nonlinear AMLI whose convergence rate is independent of problem and meshparameters. We present some numerical results for the scalar elliptic equation forhigh contrast multiscale geometries verifying our analytical findings.
69
Discontinuous Coarse Space Corrections (DCS) for OptimizedSchwarz Methods
Session Schedule Author Index Session Index M5
Kevin SantuginiInstitut Polytechnique de [email protected]
Martin J. GanderUniversite de [email protected]
Laurence HalpernUniversite Paris [email protected]
AbstractIn this presentation, we explain why continuous coarse spaces are a suboptimalchoice when combined with domain decomposition methods that have discontinuousiterates, like Optimized Schwarz Methods (OSM), or Restricted Additive Schwarzmethods (RAS). As an alternative, we propose discontinuous coarse spaces for suchdomain decomposition methods. For linear problems, we explain the design of oneparticular discontinuous coarse space and present an algorithm that computes anefficient discontinuous coarse space corrector for the special case of an OSM. Whilethe algorithm is suitable for higher dimensions, it has the special property of con-verging in a single coarse iteration for one-dimensional linear problems. We expectDiscontinuous Coarse Spaces (DCS) to become standard practice for methods withdiscontinous iterates in the coming decade.
70
Mini Symposium M6Heterogeneous Domain Decomposition Methods
Schedule Author Index Session Index
Organizers: Oliver Sander and Marco Discacciati
AbstractFor the modeling of most application problems different type of equations may berequired. This may be due either to the fact that such problems consist of more thanone single phenomenon, or that the same phenomenon is described by equations ofdifferent type in different regions of the computational domain with the aim ofreducing the overall computational cost.If the domains of definition of these equations are separated, heterogeneous domaindecomposition methods come into play. Coupling conditions between the differentmodels need to be formulated and investigated analytically. Solution algorithmsmust be constructed and their convergence properties must be studied. The sheernumber of combinations make this field abound with interesting mathematical andnumerical problems.This minisymposium aims at informing about recent developments in this field.
Part 1
Part 2
71
M6P1 Schedule
Heterogeneous Domain Decomposition Methods
Schedule Author Index Session Index
Date: Monday, June 25Time: 10:30-12:15Location: TuringChairman: Oliver Sander and Marco Discacciati
10:35-11:00 : Heiko BerningerStrategies for the Coupling of Ground and Surface WaterAbstract
11:00-11:25 : Pablo Javier BlancoCoupling Dimensionally-Heterogeneous Models in HemodynamicsSimulationsAbstract
11:25-11:50 : Eva CasoniZonal Modeling Approach in Aerodynamic SimulationAbstract
11:50-12:15 : Paola GervasioVirtual Control Method for Heterogeneous ProblemsAbstract
M6 Abstract
Part 2
72
Strategies for the Coupling of Ground and Surface Water
Session Schedule Author Index Session Index M6
Heiko BerningerUniversite de [email protected]
Ralf KornhuberFreie Universitat [email protected]
Mario OhlbergerUniversitat [email protected]
Oliver SanderFreie Universitat [email protected]
Kathrin SmetanaUniversitat [email protected]
AbstractWe discuss four different situations in which surface water is coupled to ground waterwhere the latter is considered partially saturated and modelled by the Richardsequation. The strategies for the coupling that we want to discuss involve modellingaspects as well as discretization and solution techniques. The modelling of thecoupling will be provided by mass conservation, i.e., continuity of the water flux, andeither pressure continuity or clogging effects which lead to pressure discontinuitiesacross the interface. Models for the surface water that we use are given by a waterreservoir modelled by a single ODE and including seepage faces, shallow waterequations and, finally, the ponding of water modelled by various ODE’s (one foreach point on the interface) and influenced by rainfall. The treatment of Signorini-type outflow conditions is achieved by a special discretization of the Kirchhoff-transformed Richards equation that leads to convex minimization problems. Thecoupling is treated either by time-explicit or time-implicit discretization where thelatter is solved by heterogeneous domain decomposition methods like Dirichlet-Neumann- or Robin-Neumann-type iterations. We illustrate the different couplingstrategies and models by several numerical examples.
73
Coupling Dimensionally-Heterogeneous Models inHemodynamics Simulations
Session Schedule Author Index Session Index M6
Pablo Javier BlancoLaboratorio Nacional de Computacao Cientıfica, Petropolis, [email protected]
Enzo Alberto DariComision Nacional de Energıa [email protected]
AbstractWe will present a generic black-box approach for the strong coupling of heteroge-neous flow models. The strategy is envisaged for problems arising in computationalhemodynamics, but extensions to other field of physics and engineering are straight-forward. Specifically, we will quickly revisit existing methodologies to carry out thecoupling of dimensionally-heterogeneous models, and then we will focus on tran-sient non-linear problems as those encountered when modeling the blood flow in thecardiovascular system. The proposed methodology is employed to split a coupled3D-1D-0D closed-loop model of the cardiovascular system into the correspondingblack-boxes standing for the 3D (specific vessels), 1D (systemic arteries/peripheralvessels) and 0D (venous/cardiac/pulmonary circulation) models. In addition, theacceleration of convergence in transient simulations will be discussed. We presentseveral examples of application showing the robustness and suitability of this ap-proach.
74
Zonal Modeling Approach in Aerodynamic Simulation
Session Schedule Author Index Session Index M6
Eva CasoniDep. de Matematica e Informatica, E.T.S.I Caminos, Canales y Puertos, Univ.Politecnica de [email protected]
Carlos CastroDep. de Matematica e Informatica, E.T.S.I Caminos, Canales y Puertos, Univ.Politecnica de [email protected]
AbstractZonal solvers are used to simulate CFD processes by physically partitioning a flowdomain into several regions with the aim of improving the computational overheadwhile maintaining accuracy. The main idea is to use a simpler model in particularregions in order to speed up the solver and use the full model, which is compu-tationally expensive, only where it is essential to capture the appropriate physics.In aerodynamic simulations, ignoring the viscous effects far from sharp layers leadsto the coupling of Navier-Stokes and Euler equations or Euler and full potentialequations for irrotational isentropic flows.In this talk we will focus on the analysis of two forms of the interface conditionsfor the scalar non-linear equation, also including the development of an efficientparallel strategy. Finally, their application to aerodynamic model problems thatinvolve the coupling between Euler and Navier-Stokes will be shown.
75
Virtual Control Method for Heterogeneous Problems
Session Schedule Author Index Session Index M6
Paola GervasioDepartment of Mathematics, University of Brescia, [email protected]
Marco DiscacciatiLaboratori de Calcul Numeric (LaCaN), Universitat Politecnica de Catalunya (UPCBarcelonaTech), Barcelona, [email protected]
Alfio QuarteroniMOX, Politecnico di Milano, Milano, Italy and CMCS-MATHICSE, Ecole Poly-technique Federale de Lausanne, [email protected]
AbstractIn this communication we present the Virtual Control Method (VCM) to addressheterogeneous and multiphysics problems by overlapping subdomain splitting. VCMwas proposed by Glowinski et al. in the Eighties, reconsidered by J.L. Lions andO. Pironneau at the end of the Nineties for homogeneous couplings and then bythe authors of this talk for heterogeneous problems. The basic idea of VCM con-sists in introducing suitable functions called virtual controls which play the role ofunknown boundary data on the interfaces of the decomposition and in minimizingin a suitable norm (defined on either the overlap or the interfaces) the differencebetween the two solutions defined on the same overlap region. Thus the convergencerate improves noticeably with respect to that of Schwarz method, especially whenthin overlap between subdomains is considered. Depending on the norm used inthe minimization process, the VCM can be considered to solve differential problemseither characterized by high regularity of the solution (as in homogeneous prob-lems) or, on the contrary, by a low global regularity, as it happens in heterogeneousproblems associated to multi-physics models. In this talk we consider the couplingbetween advection and advection-diffusion equations featuring boundary layers, aswell as the Stokes/Darcy coupling, with the aim of discussing both theoretical andcomputational aspects in applying VCM.
76
M6P2 Schedule
Heterogeneous Domain Decomposition Methods
Schedule Author Index Session Index
Date: Monday, June 25Time: 16:00-17:45Location: TuringChairman: Oliver Sander and Marco Discacciati
16:05-16:30 : Simona PerottoHierarchical Model Reduction: a Domain Decomposition ApproachAbstract
16:30-16:55 : Franz RammerstorferMortar FEM/BEM Coupling for PoroelastodynamicsAbstract
16:55-17:20 : Human RezaijafariA Stabilized Hybrid Discontinuous Galerkin Scheme for the NonisothermalCoupling of Stokes and Darcy FlowAbstract
17:20-17:45 : Anton SchielaEnergy Minimizers of the Coupling of a Cosserat Rod to an ElasticContinuumAbstract
M6 Abstract
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Hierarchical Model Reduction:a Domain Decomposition Approach
Session Schedule Author Index Session Index M6
Simona PerottoMOX, Dipartimento di Matematica “F. Brioschi”, Politecnico di [email protected]
AbstractIn this presentation we focus on a model reduction strategy, known as hierarchicalmodel (Hi-Mod) reduction, suited to deal with physical phenomena characterized bya dominant dynamics. As a paradigm, we can mention, for instance, flows throughporous media, flows in tubular domains (as in haemodynamics) or in a channelnetwork (as in hydrodynamics).The idea is to reduce the reference 2D or 3D (full) problem to a 1D (reduced)model associated with the leading direction, by suitably lumping the informationalong the less significant transverse directions. In particular, the reduced model islocally characterized by a different level of detail in describing the phenomenon athand according to the local relevance of the transverse dynamics. For this purpose,we resort to different discretization schemes in correspondence with the leading andthe transverse directions. The leading direction is spanned by a classical finite ele-ment scheme, while the transverse ones are expanded into a modal basis. Thus thelevel of detail of the reduced model locally varies by employing a different number ofmodal functions in different areas of the domain. A domain decomposition approachis used to enforce suitable matching conditions among the regions characterized bya different number of modes ([1, 2]).In particular, the actual goal is to devise a model-adaptive procedure, to auto-matically detect the areas as well as the number of modal functions to be locallyemployed.
References[1] A. Ern, S. Perotto and A. Veneziani, Hierarchical model reduction for advection-diffusion-reaction problems. In Numerical Mathematics and Advanced Applications,Springer-Verlag, Berlin Heidelberg, K. Kunisch, G. Of, O. Steinbach Eds. (2008),703-710. Proceedings of the 7th European Conference on Numerical Mathematicsand Advanced Applications.[2] S. Perotto, A. Ern and A. Veneziani. Hierarchical local model reduction forelliptic problems: a domain decomposition approach. Multiscale Model. Simul., 8(2010), no. 4, 1102–1127.
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Mortar FEM/BEM Coupling for Poroelastodynamics
Session Schedule Author Index Session Index M6
Franz RammerstorferInstitute of Applied Mechanics, TU Graz, [email protected]
Martin SchanzInstitute of Applied Mechanics, TU Graz, [email protected]
AbstractIn many engineering applications the numerical simulation of wave propagation phe-nomena in porous media is of great interest, e.g., in soil mechanics. In this work,aspects of time-dependent surface-coupled problems using different discretizationtechniques in different subdomains are studied. Based on Biot’s theory the gov-erning equations for the linear poroelastic continua are formulated. The couplingis done within the framework of Tearing and Interconnecting methods with finiteelements (FETI) and accordingly boundary elements (BETI). This are special non-overlapping domain decomposition methods, based on the realization of Dirichlet-to-Neumann-maps (DtN-maps) for each subdomain separately. Their representationis independent of the local discretization scheme. By means of a classical FEM dis-cretization we use a Newmark scheme in time. The discrete DtN-map for a singlesubdomain problem can be realized by eliminating the inner unknowns via Schurcomplement. In case of boundary elements, symmetric Galerkin as well as colloca-tion schemes are applicable. For time discretization the Convolution QuadratureMethod is applied. Again, using the Schur complement gives the discrete DtN-map.The continuity of the field across the interfaces is ensured by Lagrange Multipliers.Instead of the conventional node-by-node coupling this formulation is based on theMortar Method, i.e. the weak coupling is performed in terms of Lagrange multi-plier spaces. Therewith, we gain the flexibility of choosing different triangulationsfor each subdomain, which do not have coincident nodes. At the end, some nu-merical results in 3D are given to verify the presented method and to illustrate theflexibility of the algorithm.
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A Stabilized Hybrid Discontinuous Galerkin Scheme for theNonisothermal Coupling of Stokes and Darcy Flow
Session Schedule Author Index Session Index M6
Human RezaijafariM2 Zentrum Mathematik, Technische Universitat [email protected]
Barbara WohlmuthM2 Zentrum Mathematik, Technische Universitat [email protected]
AbstractThe simulation of coupled free flow and porous media flow is of special interestin many fields of application, e.g. groundwater contamination and filtration prob-lems. In this work, we consider an instationary, nonisothermal model to simulatetransport processes where the (Navier-)Stokes and Darcy equations are used to de-scribe the motion of the free and the porous media flow. The model is based ona two-domain approach and on nonisothermal compositional submodels where theemployed coupling conditions for mass, momentum and energy are based on fluxcontinuity and thermodynamic equilibrium (see [1]). Following the work of [2] wemake use of divergence-conforming Finite Elements to discretize the coupled sys-tem. We apply a mixed Discontinuous Galerkin and Finite Element method to the(Navier-)Stokes and Darcy equations, respectively. Based on the idea of having thesame error order within both subdomains we allow the use of Finite Elements of pos-sibly different order as well as nonmatching meshes at the interface. The transportand energy-balance equations are treated with a Flux-Corrected-Transport FiniteElement method (FEM-FCT, see [3]), to supress unphysical oscillations due to theconvection dominated case. We present numerical examples to verify the functional-ity of the given coupling concept with special focus on the transfer processes acrossthe interface.
References[1] K. Mosthaf, K. Baber, B. Flemisch, R. Helmig, A. Leijnse, I. Rybak and B.Wohlmuth, A new coupling concept for two-phase compositional porous media andsingle-phase compositional free flow, Water Resour. Res. 47 (2011), 1-19.[2] G. Kanschat and B. Riviere, A strongly conservative finite element method forthe coupling of Stokes and Darcy flow, J. Comput. Phys. 229 (2010) 5933-5943[3] D. Kuzmin, M. Moller and S. Turek, Multidimensional FEM-FCT schemes forarbitrary time-stepping, Int. J. Numer. Meth. 42 (2003), no 3, 265-295
80
Energy Minimizers of the Coupling of a Cosserat Rod to anElastic Continuum
Session Schedule Author Index Session Index M6
Anton SchielaTU [email protected]
Oliver SanderFU [email protected]
AbstractWe formulate the static mechanical coupling of a geometrically exact Cosserat rodto an elastic continuum. The coupling conditions accommodate for the differencein dimension between the two models. Also, the Cosserat rod model incorporatesdirector variables, which are not present in the elastic continuum model. Twoalternative coupling conditions are proposed, which correspond to two differentconfiguration trace spaces. For both we show existence of solutions of the coupledproblems. We also derive the corresponding conditions for the dual variables andinterpret them in mechanical terms.
81
82
Mini Symposium M7Domain Decomposition, Preconditioning andSolvers in Isogeometric AnalysisSchedule Author Index Session Index
Organizers: Lourenco Beirao da Veiga, Michel Bercovier, SimoneScacchi
AbstractIsogeometric Analysis (IGA) is a novel and extremely promising numerical method-ology for the analysis of PDE problems, that integrates Computer Aided Design(CAD) geometric parametrization and Finite Element analysis. Isogeometric Anal-ysis, introduced in 2005 by T.J.R. Hughes and co-workers, is having a strong impacton the engineering community, with a large amount of publications and computercodes being developed in a few years. Very recently, the research community ofthis quickly growing field has started to tackle the design of efficient solvers forIGA discrete systems, and in particular of Domain Decomposition methods yield-ing parallel and scalable IGA preconditioners. The high (global) regularity of theNURBS spaces employed in IGA discretizations introduces both new difficulties andopportunities for the construction and analysis of novel solution techniques. Theaim of the minisymposium is to bring together researchers in both fields of Isoge-ometric Analysis and Domain Decomposition, focusing on the latest developmentsand fostering new research.
Part 1
Part 2
Part 3
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M7P1 Schedule
Domain Decomposition, Preconditioning and Solvers inIsogeometric Analysis
Schedule Author Index Session Index
Date: Tuesday, June 26Time: 10:30-12:15Location: AmphiChairman: Lourenco Beirao da Veiga, Michel Bercovier, SimoneScacchi
10:35-11:00 : Remi AbgrallIsogeometric Analysis for Compressible Fluid DynamicsAbstract
11:00-11:25 : Michel BercovierIsogeometric Analysis and Schwarz Non-Matching Overlapping DomainDecomposition MethodsAbstract
11:25-11:50 : Victor M. CaloSolver Performance for Higher-Continuous BasisAbstract
11:50-12:15 : Krishan P. S. GahalautMultigrid Solver for Isogeometric DiscretizationAbstract
M7 Abstract
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Isogeometric Analysis for Compressible Fluid Dynamics
Session Schedule Author Index Session Index M7
Remi AbgrallINRIA Bordeaux Sud-Ouest and Universite de [email protected]
Cecile DobrzynskiUniversite de Bordeaux and INRIA Bordeaux [email protected]
Algiane FroehlyINRIA Bordeaux Sud-Ouest and Universite de [email protected]
AbstractDuring high order simulations, the subparametric discretization used for geo-metry’srepresentation (usually piecewise-linear) may lead to errors dominating errors re-lated to the variable field discretization. For instance, solving the conservative formof the Euler equations generate a spurious entropy that spoil the solution.We look at an isogeometric analysis aproach to solve this problem: in such methods,we use the same basis fonctions to represent the variables field and to discretize thegeometry, what ensure the same order of errors for the geometric approximationand the variables discretization.Widely used in CAO, NURBS (Non Uniform Rational B-Spline) basis functionsallow an exact representation of complicated geometries that we can meet in fluidmechanic and form an ideal family of basis functions for isogeometric ana-lysis. Alot of works coupling NURBS and finite elements method have already shown thatthe exact representation of geometric model improve significantly the numericalresults, both on quadrangular and triangular meshes. In our work, we focus on ascheme of Residual Distribution Scheme-type (RDS) which give an interesting alter-native to classical finite volume schemes: they are more accurate and their stencilis more compact.In this talk, we will first present the adaptation to isogeometric analysis of a Lax-Friedrichs-type RDS by the implementing of NURBS basis function in it. Then, wewill explain how to generate NURBS meshes and how to use mesh adaptation withNURBS meshes to reduce approximation error. At least, to illustrate the work,we will show some isogeometric numerical results for compressible fluid dynamicsand we will compared them with the results provided by the same scheme on apiecewise-linear mesh.
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Isogeometric Analysis and Schwarz Non-Matching OverlappingDomain Decomposition Methods
Session Schedule Author Index Session Index M7
Michel BercovierHebrew University of [email protected]
Ilya SoloveichikHebrew University of [email protected]
AbstractIsogeometric Analysis (IGA) is a recent technique for the discretization of PartialDifferential Equations (PDEs). One uses basis functions developed for ComputerAided Geometry for Design (CAGD) such as Non Uniform B-Splines (NURBS) todefine the same global isoparametric transformation for the ”exact” computationaldomain and for the basis functions for the PDE solution [1]. In practice objects(i.e. domains) are made up of collections of trimmed patches of CAGD transforma-tions defined through boolean operations such as union, difference or intersection( Constructive Solid Geometry:CSG). Hence it is natural to consider Domain De-composition methods as candidate solvers for ”real life’ domains. Each atom ofthe CSG construct can be considered as a simple domain, where IGA is directlyimplemented. A different application of DD to IGA is analyzed in [2].In the present work we study the simplest Schwarz Additive Domain Decomposi-tion Method (SADDM) [3] We suppose that our primitive patches are overlappingat least pair wise and that the respective isoparametric transformations are nonmatching : the pair of reference grid and knots defining each physical domain arenot related. Since at intersection the boundary of one domain is a trimming line(or surface) for its neighbor , we developed an efficient method to interpolate on atrimming line. The implementation is done then using a standard open source IGAcode for each domain, GeoPDEs [4] and illustrated on several domains in 2 and 3D.As we have shown elsewhere there is no maximum principle for IGA approximationsin 2D or 3D. Hence one needs some preconditioning to ensure convergence as wellas for large number of domains. We will discuss several types of preconditioning:such as lower degree approximations, multi-grid in relation to IGA.
[1] J.A. Cottrell,T.J.R. Hugues,Y. Bazilevs: Isogeometric Analysis ,Wiley,UK, 2009.
[2] L. Beirao da Veiga, D. Cho, L. Pavarino, S. Scacchi: Overlapping Schwarz Meth-ods for Isogeometric Analysis, preprint IMATI-CNR 8PV11/5/0, 2011.
[3] A. Toselli, O. Widlund: Domain Decomposition Methods-Algorithms and The-ory, Springer, 2004.
[4] C. de Falco, A. Reali, R. Vazquez. GeoPDEs: a research tool for IsogeometricAnalysis of PDEs. Advances in Software Engineering,40 (2011), 1020-1034.
86
Solver Performance for Higher-Continuous Basis
Session Schedule Author Index Session Index M7
Victor M. CaloKing Abdullah University of Science and Technology, Thuwal, Saudi [email protected]
David PardoThe University of the Basque Country UPV/EHU and Ikerbasque, Bilbao, [email protected]
Lisandro DalcinConsejo Nacional de Investigaciones Cientıficas y Tecnicas, Santa Fe, [email protected]
Maciej PaszynskiAGH University of Science and Technology, Krakow, [email protected]
Nathan CollierKing Abdullah University of Science and Technology, Thuwal, Saudi [email protected]
AbstractMany applications take advantage of the higher-order continuity provided by iso-geometric analysis. The approximability of higher continuous spaces per degree offreedom is superior to that of traditional finite element spaces. This suggests thatisogeometric analysis links geometry to analysis and is a more efficient method.However, the connection between the number of degrees of freedom and the or-der of approximation is not a measure of efficiency. We also need to consider thecost associated with solving the linear system generated by the discrete weak form.The system formed by higher continuous basis has a denser structure than thatof standard finite element spaces. The support of more continuous functions ex-tends beyond element boundaries resulting in a more connected matrix. The moreconnected matrix degrades the performance of direct solver. Similarly for iterativesolvers, the matrices resulting from higher-continuous basis functions contain morenon-zero entries. The number of non-zero entries directly relates to the cost ofthe matrix-vector multiplication, a major contribution to the cost of the iterativemethod. We will present estimates for the increase in cost. However, developingcost estimates for iterative solvers is far more complex than in direct solvers. Whilethe cost per iteration increases, the number of iterations is also an important com-ponent. Thus, our analysis includes estimates for the iteration count required forconvergence using conjugate gradients with some popular preconditioners. Our nu-merical results indicate how to construct optimal preconditioners for the Laplaceproblem.
87
Multigrid Solver for Isogeometric Discretization
Session Schedule Author Index Session Index M7
Krishan GahalautJohann Radon Institute for Computational and Applied Mathematics (RICAM)[email protected]
Johannes KrausJohann Radon Institute for Computational and Applied Mathematics (RICAM)[email protected]
Satyendra TomarJohann Radon Institute for Computational and Applied Mathematics (RICAM)[email protected]
AbstractWe present the (geometric) multigrid methods for the isogeometric discretizationof Poisson equation. The smoothing property of the relaxation method, and theapproximation property of the intergrid transfer operators are analyzed for two-grid and multi-grid cycles. It is shown that the convergence of the multigrid solveris independent of the discretization parameter h, and that the overall solver is ofoptimal complexity. Supporting numerical results are provided for the smoothingproperty, the approximation property, convergence factor and iterations count forV -, W - and F - cycles, and the linear dependence of V -cycle convergence on thesmoothing steps. The numerical results are complete up to polynomial degree p = 4,and for minimum smoothness C0 and maximum smoothness Cp−1.
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M7P2 Schedule
Domain Decomposition, Preconditioning and Solvers inIsogeometric Analysis
Schedule Author Index Session Index
Date: Tuesday, June 26Time: 14:45-15:35Location: AmphiChairman: Lourenco Beirao da Veiga, Michel Bercovier, SimoneScacchi
14:45-15:10 : Christian HeschMortar Based Domain Decomposition for Isogeometric AnalysisAbstract
15:10-15h35 : Stefan KleissIETI - Isogeometric Tearing and InterconnectingAbstract
M7 Abstract
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89
Mortar Based Domain Decomposition for Isogeometric Analysis
Session Schedule Author Index Session Index M7
Christian HeschUniversity of [email protected]
Peter BetschUniversity of [email protected]
AbstractIn the present talk we present a nonconform domain decomposition method basedon mortar methods in the context of isogeometric analysis. The use of NURBS asbasis for Galerkin-based discretization schemes within the field of nonlinear elastic-ity necessitates the development of a versatile interface between sub-domains whichare independently h-, p- or k-refined. Additionally it is of interest to provide anadequate coupling mechanism for sub-domains, discretized either with Lagrangianor NURBS based shape functions. In particular, we apply a variant of the mortarmethod using Lagrangian based shape functions for the interpolation of the La-grange multiplier field, evaluated at the discrete surface of the master side. Sincethe control points of the NURBS may, but do not have to be part of the geometry,we can not use them directly for the tying of the dissimilar meshes, although thesecontrol points are our primarily variables. Here, we take advantage of the fact thatNURBS of arbitrary order are always partitioned by knot-vectors, which gives usa similar finite element structure as for Lagrangian meshes, and apply the linearinterpolation of the Lagrange multiplier field directly to the elements of the surfaceof the master side. At last, well-known concepts to enhance numerical stability forthe discretization in time can also been applied to both, the mechanical field aswell as to the interface constraints. Those energy-momentum conserving schemesfacilitate algorithmic conservation of total energy as well as both momentum mapsand can be applied without modification to isogeometric problems.
90
IETI – Isogeometric Tearing and Interconnecting
Session Schedule Author Index Session Index M7
Stefan KleissJohann Radon Institute for Computational and Applied Mathematics, Linz, Aus-trian Academy of Sciences (OAW)[email protected]
Clemens PechsteinInstitute of Computational Mathematics, Johannes Kepler University [email protected]
Bert JuttlerInstitute of Applied Geometry, Johannes Kepler University [email protected]
Satyendra TomarJohann Radon Institute for Computational and Applied Mathematics, Linz, Aus-trian Academy of Sciences (OAW)[email protected]
AbstractFinite Element Tearing and Interconnecting (FETI) methods are a powerful ap-proach to designing solvers for large-scale problems in computational mechanics.The numerical simulation problem is subdivided into a number of independent sub-problems, which are then coupled in appropriate ways. NURBS- (Non-Uniform Ra-tional B-spline) based isogeometric analysis (IGA) applied to complex geometriesrequires to represent the computational domain as a collection of several NURBSgeometries. Since there is a natural decomposition of the computational domaininto several subdomains, NURBS-based IGA is particularly well suited for usingFETI methods.
We propose the new IsogEometric Tearing and Interconnecting (IETI) method,which combines the advanced solver design of FETI with the exact geometry repre-sentation of IGA. We describe the IETI framework for two classes of simple modelproblems (Poisson and linearized elasticity) and discuss the coupling of the subdo-mains along interfaces (both for matching interfaces and for interfaces with T-joints,i.e. hanging knots). Attention is also paid to the construction of a suitable precondi-tioner for the iterative linear solver used for the interface problem. We report severalcomputational experiments to demonstrate the performance of the proposed IETImethod.
91
M7P3 Schedule
Domain Decomposition, Preconditioning and Solvers inIsogeometric Analysis
Schedule Author Index Session Index
Date: Tuesday, June 26Time: 16:00-17:45Location: AmphiChairman: Lourenco Beirao da Veiga, Michel Bercovier, SimoneScacchi
16:05-16:30 : Angela KunothMultilevel Preconditioning for Isogeometric AnalysisAbstract
16:30-16:55 : Luca F. PavarinoOverlapping Schwarz Methods for Isogeometric AnalysisAbstract
16:55-17:20 : Satyendra TomarAlgebraic Multilevel Iteration Method for Isogeometric Discretization ofElliptic ProblemsAbstract
17:20-17:45 : Rafael VazquezMultilevel Preconditioning for Isogeometric Analysis Based on HierarchicalSplinesAbstract
M7 Abstract
Part 1
Part 2
92
Multilevel Preconditioning for Isogeometric Analysis
Session Schedule Author Index Session Index M7
Angela KunothInstitut fur Mathematik, Universitat Paderborn, Germany,www2.math.uni-paderborn.de/ags/kunoth/group/[email protected]
AbstractWe consider elliptic PDEs (partial differential equations) in the framework of iso-geometric analysis, i.e., we treat the physical domain by means of a B-spline orNurbs mapping which we assume to be regular. The numerical solution of thePDE is based on tensor products of B-splines of degree p on uniform grids of gridspacing h. We construct additive multilevel preconditioners and show that they areasymptotically optimal, i.e., the spectral condition number of the resulting stiffnessmatrix is independent of h. Together with a nested iteration scheme, this enables aniterative solution scheme of optimal linear complexity. The theoretical results aresubstantiated by numerical examples in up to three space dimensions for differentdegrees p. This is joint work with Annalisa Buffa, Helmut Harbrecht, GiancarloSangalli and others.
93
Overlapping Schwarz Methods for Isogeometric Analysis
Session Schedule Author Index Session Index M7
Luca Franco PavarinoUniversita di Milano, [email protected]
Lourenco Beirao da VeigaUniversita di Milano, [email protected]
Durkbin ChoDongguk University, Seoul, South [email protected]
Simone ScacchiUniversita di Milano, [email protected]
AbstractWe construct and analyze an overlapping Schwarz preconditioner for elliptic prob-lems discretized with NURBS-based isogeometric analysis. We consider both scalarproblems and the system of linear elasticity, in two and three dimensions. Thepreconditioner is based on partitioning the domain of the problem into overlappingsubdomains, solving local isogeometric problems on these subdomains and solvingan additional coarse isogeometric problem associated with the subdomain mesh. Wedevelop an h-analysis of the preconditioner, showing in particular that the resultingalgorithm is scalable and its convergence rate depends linearly on the ratio betweensubdomain and “overlap sizes”, for fixed polynomial degree p and regularity k of thebasis functions. Numerical results in 2D and 3D tests show the good convergenceproperties of the preconditioner with respect to the isogeometric discretization pa-rameters h, p, k, number of subdomains N , overlap size and also with respect tojumps in the coefficients of the elliptic operator.
References:
L. Beirao da Veiga, D. Cho, L. F. Pavarino, S. Scacchi. Overlapping Schwarz meth-ods for Isogeometric Analysis. SIAM J. Numer. Anal., 2012
94
Algebraic Multilevel Iteration Method for IsogeometricDiscretization of Elliptic Problems
Session Schedule Author Index Session Index M7
Satyendra TomarRICAM, Austrian Academy of Sciences, Linz, [email protected]
Krishan GahalautRICAM, Austrian Academy of Sciences, Linz, [email protected]
Johannes KrausRICAM, Austrian Academy of Sciences, Linz, [email protected]
AbstractIn this talk we shall present an algebraic multilevel iteration method for solving lin-ear systems arising from the isogeometric discretization of elliptic boundary valueproblems. Theoretical bounds for the constant γ in the strengthened Cauchy-Bunyakowski-Schwarz inequality will be discussed. Some numerical results, sup-porting the theoretical estimates, will be presented. A comparison with recentlyintroduced multigrid methods for isogeometric discretization [1] will also be drawn.
[1] K.P.S. Gahalaut, J.K. Kraus and S.K. Tomar: Multigrid Methods for Isogeo-metric Discretization. Submitted for publication. Also available as RICAM Report2012-08.
95
Multilevel Preconditioning for Isogeometric Analysis Based onHierarchical Splines
Session Schedule Author Index Session Index M7
Rafael VazquezIstituto di Matematica Applicata e Tecnologie Informatiche del CNR, [email protected]
Annalisa BuffaIstituto di Matematica Applicata e Tecnologie Informatiche del CNR, [email protected]
Helmut HarbrechtMathematisches Institut, Universitat [email protected]
Angela KunothInstitut fur Mathematik, Universitat [email protected]
Giancarlo SangalliDipartimento di Matematica, Universita di [email protected]
AbstractIsogeometric analysis (IGA) is a non-standard technique for the discretization ofPDEs, which basically consists on approximating the solution of the equations withthe same functions that describe the geometry in CAD, such as NURBS or B-splines.The drawback of these functions is their tensor product structure, that forces therefinement to propagate to all the domain. Some alternative functions that allowfor local refinement, such as T-splines and LR-splines, have been already proposedas a tool for IGA. Also hierarchical splines, that are already well known in the CADcommunity, were studied in the isogeometric context. Like T-splines, hierarchicalsplines can be refined locally maintaining high interelement continuity, but with amuch easier construction and implementation. The advantage of this is that theycan be easily defined also in the three-dimensional case and for arbitrary degrees.
In this work we present a construction of hierarchical splines that defines the setof basis functions in a truly hierarchical way. This approach gives a canonical con-struction of hierarchical splines, avoiding some of the difficulties present in previousworks, such as the necessity of removing linearly dependent functions. At the sametime, it automatically provides all the ingredients for the construction of multilevelpreconditioners. We will show preliminary numerical results of the performance ofthe BPX preconditioner using hierarchical splines.
96
Mini Symposium M8Domain Decomposition Techniques in Practi-cal Flow ApplicationsSchedule Author Index Session Index
Organizers: Menno Genseberger, Mart Borsboom, Martin J. Gander
AbstractThe last decades domain decomposition techniques have been incorporated in largecomputer codes for real life applications. This minisymposium brings together someof them, here the aim is twofold. On one hand the organizers want to illustrate theimportance of domain decomposition (for instance for modelling flexibility or paral-lel performance) in the application field. On the other hand the intention is to high-light the applied domain decomposition techniques, to discuss these approaches and-if needed- reconsider or further improve them. The application area is restricted tohydrodynamics, as this will yield a good basis for further discussion. The presenta-tions consider domain decomposition techniques in large computer codes that arebeing used world wide for shallow water flow in coastal areas, lakes, rivers, oceanflow and climate modelling.
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M8 Schedule
Domain Decomposition Techniques in Practical FlowApplications
Schedule Author Index Session Index
Date: Monday, June 25Time: 16:00-17:45Location: I50Chairman: Menno Genseberger, Mart Borsboom, Martin J. Gander
16:05-16:30 : Eric BlayoInterface Conditions and Domain Decomposition Methods forOcean-Atmosphere CouplingAbstract
16:30-16:55 : Bas van ’t HofWater Level Predictions with WAQUA: Domain Decomposition on a DailyBasisAbstract
16:55-17:20 : Mart BorsboomAnalysis and Optimization of the Coupling Between Non-OverlappingSubdomains in 1DAbstract
17:20-17:45 : Fred WubsHYMLS: a Robust Parallel Preconditioner for Fluid Flow ComputationsAbstract
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Interface Conditions and Domain Decomposition Methods forOcean-Atmosphere Coupling
Session Schedule Author Index Session Index M8
Eric BlayoLaboratoire Jean Kuntzmann, University of [email protected]
Florian LemarieCenter for Earth Systems Research, University of California at Los [email protected]
Laurent DebreuLaboratoire Jean Kuntzmann, INRIA, [email protected]
AbstractThe interactions between atmosphere and ocean play a major role in many geophysi-cal phenomena, distributed over a wide range of temporal scales (e.g. breeze diurnalcycle, tropical cyclones, global climate dynamics...). Therefore the numerical simu-lation of such phenomena require atmospheric and oceanic coupled models, whichproperly represent the behavior of the boundary layers encompassing the air-seainterface and their two-way interactions. Early studies making use of such coupledmodels often have difficulties in capturing themesoscale air-sea interaction, thusleading to a very weak coupling generally inconsistent with available observations.Deficiencies appear both in the formulation of the physical parameterizations andin the algorithmic approach used for thecoupling. In this talk, we will address thisproblem from the point of view of domain decomposition methods. We will showthat present coupling methods used for ocean-atmosphere coupled models can bewritten in the formalism of Schwarz iterative algorithms, and correspond to methodswhich are not pushed to convergence, and may lead to quite imperfect coupling. Inparticular, atmosphere and ocean coupled solutions may exhibit a strong sensitivityto model parameters and can be inherently uncertain. We will show that using im-proved coupling algorithms (like Schwarz methods) can reduce this sensitivity quitesignificantly, and we will discuss the overall objective of achieving a mathematicallyandphysically consistent ocean-atmosphere coupling.
99
Water Level Predictions with WAQUA:Domain Decomposition on a Daily Basis
Session Schedule Author Index Session Index M8
Bas van ’t HofVORtech [email protected]
AbstractFor over fifteen years, the Dutch Ministry for Infrastructure and the Environmenthas been working on domain decomposition techniques for an application calledWAQUA. My presentation will cover the following aspects of domain decomposi-tion in WAQUA.
WAQUA is an application for 2D and 3D simulation of flow and transport in estuar-ies, coastal waters and rivers. It is a consultancy tool, used for predicting the effectsof (infrastructural) changes, and is also used in operational systems which guard thecountry’s safety from floods. Especially in operational systems, it is a unique prod-uct, which predicts water levels along the Dutch coast very accurately. WAQUA isused by the Dutch Meteorological Institute (KNMI), Storm Surge Warning Service(SVSD) and Hydro Meteo Centra (HMCs) and others.
The domain decomposition approach for WAQUA, developed by VORtech Com-puting, has evolved from a method for parallel computing into a tool for increasedmodelling flexibility, because every domain in the simulation may have its own gridand physical model.
WAQUA’s use for consultancy and in vital operational systems creates strong re-quirements on usability, robustness and computational efficiency of the softwareand the domain decomposition functionality. Usability requirements have inspired(graphical) tools for coupling the domains, as well as pre- and postprocessing facil-ities which allow the user to interpret the input and output of different domains inas simple and intuitive a way as possible.
The computational requirements (robustness and efficiency) are met using a power-ful system for information exchange between different domains, allowing program-mers of numerical algorithms to specify very exactly and very compactly which in-formation to exchange at which times. This allows numerical aspects of distributedcalculations to be separated from the technical aspects of data exchange.
This powerful data-exchange support is illustrated by the distributed algorithmused in the convection-diffusion equation: a simple red-black Jacobi iteration. Aninteresting case is the solution of the coupled momentum and continuity equations,a large number of distributed, one-dimensional, nonlinear systems. The solutionof these systems requires a generalization of the efficient Block Jacobi Two WayGaussian Elimination for cases with grid refinements.
100
Analysis and Optimization of the Coupling BetweenNon-Overlapping Subdomains in 1D
Session Schedule Author Index Session Index M8
Mart [email protected]
AbstractBesides using domain decomposition techniques to partition problems for parallelcomputation, some of these techniques can also be used to partition problem ar-eas in subdomains that each may be modeled differently. It would be useful tohave a multi-domain simulation environment where each subdomain could have itsown numerical approach, i.e., its own computational grid, time step, discretizationmethod, and iterative solver. Allowing different model equations per subdomain(describing the same physical process of course) would further enhance modelingflexibility. For example, an elaborate flow model, a high-order accurate discretiza-tion, a fine grid, and a small time step may only be required in areas where thedetailed LES simulation of turbulent flow is important, e.g., near a structure forthe prediction of forces. Elsewhere a simpler and cheaper modeling approach maysuffice.A multi-domain modeling framework where computational models can be set up in-dependently per subdomain requires couplings at the subdomain interfaces that aregenerally applicable and do not introduce any significant errors. Suitable methodsand techniques need to be developed to realize such general-purpose interfaces atthe physical level, at the numerical level, as well as at the software/hardware level.This talk will address some of the numerical issues associated with subdomain in-terfaces. By considering the 1D linear shallow-water equations, an analysis can beperformed on the space discretizations applied inside the subdomains and at theinterface between them, on the spurious effects that this may cause, and on the min-imization of these errors by optimization of the schemes applied at the interface.As for the integration in time, general applicability (and parallelization as well)requires the coupling between subdomains to be explicit. This poses no problem ifan explicit time integration scheme is applied. However, if the scheme is (partially)implicit, iterations across the subdomains are generally required. We will showthat by applying high-order extrapolations in time or other explicit techniques atthe interface, highly improved yet stable initial solution estimates can be obtainedthat will strongly accelerate that iteration process.An optimized general-purpose DD coupling based on 1D linear theory is essentialand a good start, but nonlinear and multi-dimensional directional effects need tobe included as well. We will comment on these issues at the presentation.
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HYMLS: a Robust Parallel Preconditioner for Fluid FlowComputations
Session Schedule Author Index Session Index M8
Fred WubsJohann Bernoulli Institute for Mathematics and Computing Science, University ofGroningen, The NetherlandsEmail: [email protected]
Jonas ThiesCentre for Interdisciplineary Mathematics, Uppsala University, SwedenEmail: [email protected]
AbstractIn this contribution we discuss a new hybrid direct/iterative approach to the solutionof a special class of saddle point matrices arising from the discretization of thesteady (Navier-) Stokes equations on an Arakawa C-grid, F-matrices. In generalsuch matrices come about from a finite volume discretization (Marker and Cell)on a structured grid as is common for incompressible flows. The two-level methodintroduced here [2] is derived from a direct method for the same type of problems [1]and has the following properties: (i) it is very robust, even close to the point wherethe solution becomes unstable; (ii) a single parameter controls fill and convergence,making the method straightforward to use; (iii) the convergence rate is independentof the number of unknowns; (iv) it can be implemented on distributed memorymachines in a natural way [3]; (v) the matrix on the second level has the samestructure and numerical properties as the original problem, so the method can beapplied recursively; (vi) the iteration takes place in the divergence-free space, sothe method qualifies as a ‘constraint preconditioner’; (vii) the approach can also beapplied to a number of simpler problems like the Poisson and Darcy equations. Inthe talk we will give an outline of the method and show its parallel performance onthe incompressible Navier-Stokes equations.
[1] A.C. de Niet and F.W. Wubs. Numerically stable LDLT -factorization of F-typesaddle point matrices. IMA J. Numer. Anal., 29:208234, 2009.
[2] Fred W. Wubs and Jonas Thies. A Robust Two-Level Incomplete Factorizationfor (Navier)Stokes Saddle Point Matrices. SIAM. J. Matrix Anal. and Appl. 32:1475–1499, 2011.
[3] Jonas Thies and Fred Wubs. Design of a Parallel Hybrid Direct/Iterative Solverfor CFD Problems. Proceedings of the 2011 IEEE 7th International Conference onE-Science, pp. 387–394, 2011.
102
Mini Symposium M9Fast Solvers for Helmholtz and Maxwell equa-tionsSchedule Author Index Session Index
Organizers: Victorita Dolean, Ronan Perrussel, Hui Zhang, Peng Zhen
AbstractWave propagation problems are intrinsically very challenging from mathematicalpoint of view especially in the time-harmonic regime. The major difficulties stemfrom the mathematical models (Helmholtz or Maxwell equations) which are in-definite Partial Differential Equations treating phenomena of ondulatory nature.During the last decade, motivated by a large panel of applications, approximationand solution methods for these equations, arose the interest of a largePof scientificcommunity. In this mini-symposium we propose an overview of different techniqueswhich lead to fast solutions of Helhmholtz and Maxwell equations: optimized trans-mission conditions, discrete frameworks of domain decomposition methods, couplingof different discretizations in the domain decomposition spirit or the treatement ofheterogeneous problems. The mini-symposium is organized in four sessions as fol-lows: in the first session, chaired by Hui Zhang, after giving an overview of differentsolvers (speaker 1), different ways of improving the convergence of domain decompo-sition algorithms applied to Helmholtz equations by analythically inspired methodsare shown (speakers 2-4). The second session chaired by Peng Zhen (speakers 5-8) isabout coupling discretizations or alternative solvers/preconditioners for Helmholtzand Maxwell’s equations. The third one, chaired by Victorita Dolean (speakers9-12), treats the topic of discrete frameworks leading to efficient solvers. The lastsession, chaired by Ronan Perrussel (speakers 13-16) mainly oriented on Maxwell’sequations, treats about efficient computations and implementations of domain de-composition methods.
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Part 2
Part 3
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M9P1 Schedule
Fast Solvers for Helmholtz and Maxwell equations
Schedule Author Index Session Index
Date: Tuesday, June 26Time: 10:30-12:15Location: TuringChairman: Victorita Dolean, Ronan Perrussel, Hui Zhang, Peng Zhen
10:35-11:00 : Lea ConenAn Overview of Multigrid and Domain Decomposition Methods for theHelmholtz EquationAbstract
11:00-11:25 : Hui ZhangOptimized Schwarz Methods with Overlap for Helmholtz EquationAbstract
11:25-11:50 : Erwin VenerosOptimized Schwarz Methods for Maxwell Equations with DiscontinousCoefficientsAbstract
11:50-12:15 : Bertrand ThierryImproved Domain Decomposition Method for the Helmholtz EquationAbstract
M9 Abstract
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Part 3
Part 4
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An Overview of Multigrid and Domain Decomposition Methodsfor the Helmholtz Equation
Session Schedule Author Index Session Index M9
Lea ConenInstitute of Computational Science, University of Lugano, Lugano, [email protected]
Rolf KrauseInstitute of Computational Science, University of Lugano, Lugano, [email protected]
AbstractThe Helmholtz equation describes linear, time-harmonic wave problems that arisein many engineering fields such as aeronautics and electromagnetic applications.The wave-like behavior of the solution and the indefinite, severely ill-conditionedlinear systems resulting from its discretization by means of finite elements make itsnumerical solution difficult. Special care is needed to cope with these peculiarities asstandard iterative methods such as multigrid are not robust for Helmholtz problems.Therefore, during the last decades, a large variety of suitably adapted iterativemethods has been proposed aiming at the development of efficient and fast solutionstrategies.In this talk, we give an overview of multigrid and domain decomposition methodsfor the solution of the systems stemming from finite element discretizations of theHelmholtz equation. In particular, the definition of a coarse space capturing thecharacteristics of the problem and the definition of suitable transmission conditions– between subdomains in the domain decomposition and between grid levels in themultigrid context – are crucial. We present techniques that have been developedand successfully applied by different authors in the course of the last years anddiscuss differences to the positive definite, elliptic case.
105
Optimized Schwarz Methods with Overlap for the HelmholtzEquation
Session Schedule Author Index Session Index M9
Hui ZhangUniversity of [email protected]
Martin J. GanderUniversity of [email protected]
AbstractOptimized Schwarz methods without overlap for the Helmholtz equation were pro-posed in Gander, Magoules, Nataf, 2002 and further developed in Gander, Halpern,Magoules, 2007. We will present here optimized Schwarz methods with overlap forthe Helmholtz equation. By scaling the mesh size and the wave-number, and usingasymptotic analysis, we are able to give easy to use formulas for calculating theoptimized parameters of the method. The convergence rates are also derived fromthe corresponding asymptotic analysis, and we can show an important improve-ment, due to the overlap. We also illustrate our theoretical results with numericalexperiments.
106
Optimized Schwarz Methods for Maxwell Equations withDiscontinous Coefficients
Session Schedule Author Index Session Index M9
Erwin VenerosUniversite de [email protected]
Victorita DoleanUniversite de Nice and Universite de [email protected]
Martin J. GanderUniversite de [email protected]
AbstractWe study non-overlapping Schwarz Methods for solving time-harmonic Maxwell’sequations in heterogeneous media. For this paper we consider Maxwell’s equa-tions in two dimensions in both the transverse electric and transverse magneticmode formulations. We first present the classical Schwarz Method for the problem,which uses characteristic transmission conditions and can therefore be used withoutoverlap. Choosing the interfaces between the subdomains aligned with the disconti-nuities in the coefficients, we prove convergence of the method for a model problem.We then define several optimized transmission conditions dependent on the disconti-nuities of the magnetic permeability and the electric permittivity. These conditionsare determined by solving the corresponding min-max problems. We prove asymp-totically that the resulting methods converge in certain cases independently of themesh parameter, even though the methods are non-overlapping. We illustrate ourtheoretical results with numerical experiments.
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Improved Domain Decomposition Method for the HelmholtzEquation
Session Schedule Author Index Session Index M9
Bertrand ThierryMontefiore Institut, University of Liege, [email protected]
Xavier AntoineInstitut Elie Cartan de Nancy, University of Nancy, [email protected]
Yassine BoubendirNew Jersey Institut of Technology, [email protected]
Christophe GeuzaineMontefiore Institut, University of Liege, [email protected]
Alexandre VionMontefiore Institut, University of Liege, [email protected]
AbstractIn this talk we will present recent improvements to the quasi-optimal domain de-composition method for the Helmholtz equation presented in [1]. The key pointof the method is the construction of an accurate local approximation of the exactDirichlet-to-Neumann operator which leads to a new transmission operator betweensub-domains. We will show that this local approximation, based on complex Padapproximants, is well-suited for large scale parallel finite element simulations of highfrequency scattering problems, with either manual or automatic mesh partitioning.In particular, we will show that our algorithm is quasi-optimal in the sense that theconvergence rate of the iterative solver depends only slightly on both the frequencyand the mesh refinement.
[1] Y. Boubendir, X. Antoine and C. Geuzaine, A Quasi-Optimal Non-OverlappingDomain Decomposition Algorithm for the Helmholtz Equation. Journal of Compu-tational Physics 231 (2), (2012), pp.262-280
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M9P2 Schedule
Fast Solvers for Helmholtz and Maxwell equations
Schedule Author Index Session Index
Date: Tuesday, June 26Time: 16:00-17:45Location: TuringChairman: Victorita Dolean, Ronan Perrussel, Hui Zhang, Peng Zhen
16:05-16:30 : Olaf SteinbachCoupled Finite and Boundary Element Methods for Vibro-Acoustic InterfaceProblemsAbstract
16:30-16:55 : Jin-Fa LeeIntegral Equation Domain Decomposition Method for SolvingElectromagnetic Wave Scattering from Deep CavitiesAbstract
16:55-17:20 : Eric DarrigrandOSRC Preconditioner and Fast Multipole Method for 3D HelmholtzEquation: a Spectral AnalysisAbstract
17:20-17:45 : Yogi ErlanggaShift-Operator-Based Domain Decomposition Method for the HelmholtzEquationAbstract
M9 Abstract
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Coupled Finite and Boundary Element Methods forVibro-Acoustic Interface Problems
Session Schedule Author Index Session Index M9
Olaf SteinbachTU [email protected]
AbstractFor the vibro–acoustic simulation of completely immersed bodies such as submarineswe discuss coupled finite and boundary element formulations in the frequency do-main. While the vibro–elastic material in a bounded domain is handled by a stan-dard finite element approach, the acoustic fluid in the unbounded exterior domain isincorporated via boundary integral equations. As well known for boundary integralequations related to the Helmholtz equation, special care is required to avoid spuri-ous modes. We discuss several formulations which all result in stable discretizationschemes, and we discuss appropriate solution methods for the direct problem. Todetermine critical frequencies for the coupled problem we discuss related eigenvalueproblems. When considering a simplified Laplace model in the exterior domain, theeigenvalue problem is linear in the wave number, while for the Helmholtz model wehave to solve an eigenvalue problem which depends nonlinear on the wave number.We present both analytic and numerical results for such classes of coupled problems.The talk is based on joint work with G. Unger and A. Kimeswenger.
110
Integral Equation Domain Decomposition Method for SolvingElectromagnetic Wave Scattering from Deep Cavities
Session Schedule Author Index Session Index M9
Jin-Fa LeeElectroScience Laboratory, The Ohio State [email protected]
Zhen PengElectroScience Laboratory, The Ohio State [email protected]
AbstractWe introduce a novel boundary integral equation domain decomposition methodfor solving time harmonic electromagnetic wave scattering from a large and deepinlet with lossy thin coating, which is embedded in an arbitrarily shaped host body.The proposed method follows a hierarchical domain partitioning strategy. First,in order to decompose the entire problem domain into the interior cavity and theexterior host body regions, an artificial surface is placed over the opening of theinlet, as a transmission interface. Subsequently, a new generalized combined fieldintegral equation (G-CFIE), whose unknowns include surface traces of both electricand magnetic fields, is employed for both the interior and the exterior sub-domains.Finally, the required couplings between the interior cavity and the exterior hostbody are then completely channeled through the artificial interface surface, via theapplication of a Robin transmission condition. Additionally, the interior cavityregion can be further partitioned into many sub-domains, again through artificialsurfaces across the inlet aperture. In doing so, the dense couplings in tradition-ally integral equations are effectively transformed into sparse couplings as in finitemethods such as finite element methods. Our treatments of the interior cavity re-gion indeed bear many similarities to the popular boundary element tearing andinterconnecting (BETI) methodology. The strength and flexibility of the proposedmethod will be distinctly illustrated by means of several representative numericalexamples.
111
OSRC Preconditioner and Fast Multipole Method for 3-DHelmholtz Equation: a Spectral Analysis
Session Schedule Author Index Session Index M9
Eric DarrigrandIRMAR – Universite de Rennes [email protected]
Marion DarbasLAMFA – Universite de [email protected]
Yvon LafrancheIRMAR – Universite de Rennes [email protected]
AbstractIn acoustic scattering, integral equation methods are widely used to study wavepropagation outside a bounded 3-D domain. Thanks to these techniques, the gov-erning boundary-value problem is reduced to an integral equation on the surface Γof the scatterer ([Colton-Kress, 83]). We consider here impenetrable bodies withsmooth boundaries and incident time-harmonic plane waves. The Combined FieldIntegral Equation (CFIE) is uniquely solvable in H1/2(Γ) for all frequency k > 0,and implies the first and second traces of the double-layer potential respectivelydenoted M and D. In terms of numerical iterative resolution, this equation doesnot provide a good spectral behavior due to the strongly singular and non-compactoperator D. A strategy consists in preconditioning the operator D by introducingan efficient approximation V of the exterior Neumann-to-Dirichlet (NtD) map, us-ing On-Surface Radiation Condition (OSRC) methods ([Antoine-Darbas, 07]). Thepreconditioned equation is uniquely solvable for all wavenumbers and exhibits veryinteresting spectral properties. The OSRC technique only involves local operators,so that the numerical implementation of the preconditioning operator requires onlythe use of a sparse direct solver and does not really affect the cost of the itera-tive resolution of the CFIE equation. The most expensive part of the resolutionis still consequent to the integral operators. The Fast Multipole Method (FMM)([Coifman-Rokhlin-Wandzura, 93]) is then considered to deal with the operators Mand D. A thorough study of the eigenvalues behavior is realized in order to illustratethe impact of the OSRC-preconditioning technique on the spectrum of the CFIEoperator. The resolution scheme is applied to several numerical test-cases (sphere,cube, trapping domain). The convergence of the GMRES corroborates the spectralanalysis. Only a few GMRES iterations are required for both high frequencies andrefined meshes. The computation cost follows the FMM behavior. Combining theOSRC preconditioner and the FMM proves to be a very efficient approach to solvethe CFIE at high frequencies.
112
Shift Operator-Based Domain Decomposition Method for theHelmholtz Equation
Session Schedule Author Index Session Index M9
Yogi Ahmad ErlanggaCollege of Science, Alfaisal University, Riyadh, [email protected]
AbstractThe publication of [1], has revived research on fast iterative solvers for the Helmholtzequation. While it is believed that the Helmholtz equation still poses problems forclassical iterative solvers [3], the role of complex (shifted) Laplacian as precondi-tioner has been recognized very well in handling difficulties due to indefiniteness ofthe Helmholtz matrix. In general, the convergence of a Krylov method, precondi-tioned by the shifted Laplacian, behaves linearly with respect to the wavenumber,with a small constant. In a rather obscure paper [2], the authors incorporate asecond-level preconditioner to tackle the small eigenvalues based on a modified de-flation (shift) operator, which successfully shifts small eigenvalues towards the upperbound. Numerical tests revealed an almost wavenumber-independent convergence,the first time ever observed.The shift operator mimics both domain-decomposition and multigrid. While it iseasier to implement the operator in a multigrid fashion, it is interesting to see ifthis operator can be cast into a domain-decomposition method, event though notin the classical sense. This will somehow enlarge the deflation subspace and providea more efficient means to handle the shift operator via subdomain solves. Whilefor the Poisson problem, this is already the case, for the Helmholtz equation, thisis feasible but the implementation is not straightforward.In this talk, we will discuss and show what has been achieved in this direction,based on 1D and 2D Helmholtz problems.
[1] Y.A. Erlangga, C.W. Oosterlee, and C. Vuik, A novel multigrid-based precondi-tioner for the heterogeneous Helmholtz equation, SIAM Journal on Scientific Com-puting, 27 (2006), pp. 1471–1492.
[2] Y.A. Erlangga and R. Nabben, On a multilevel Krylov method for the Helmholtzequation preconditioned by shifted Laplacian, Electronic Transaction on NumericalAnalysis, 2008 (31) pp. 203–234.
[3] O.G. Ernst and M.J. Gander, Why it is difficult to solve Helmholtz problemswith classical iterative methods, in Numerical Analysis of Multiscale Problems, I.Graham, T. Hou, O. Lakkis and R. Scheichl, Editors, Springer Verlag, 2011.
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M9P3 Schedule
Fast Solvers for Helmholtz and Maxwell equations
Schedule Author Index Session Index
Date: Wednesday, June 27Time: 10:30-12:15Location: TuringChairman: Victorita Dolean, Ronan Perrussel, Hui Zhang, Peng Zhen
10:35-11:00 : Rosalie Belanger-RiouxA Fast and Accurate Absorbing Boundary Condition for the HelmholtzEquationAbstract
11:00-11:25 : Achim SchadleCurl-Conforming Hardy Space Infinite Elements for Exterior MaxwellProblemsAbstract
11:25-11:50 : Ana Alonso RodriguezFinite Element Construction of Discrete Harmonic FieldsAbstract
11:50-12:15 : Martin HuberHybrid Domain Decomposition Solvers for the Helmholtz EquationAbstract
M9 Abstract
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A Fast and Accurate Absorbing Boundary Condition for theHelmholtz Equation
Session Schedule Author Index Session Index M9
Rosalie Belanger-RiouxDepartment of Mathematics, Massachusetts Institute of [email protected]
Laurent DemanetDepartment of Mathematics, Massachusetts Institute of [email protected]
AbstractConstructing accurate absorbing, radiating or non-reflecting boundary conditions(hereafter ABCs) for the Helmholtz equation
∆u(x) +ω2
c2(x)u(x) = f(x)
in heterogeneous media (i.e. c not constant) is difficult and costly. In particu-lar, there are a number of applications, such as imaging, where is it necessary tosolve this equation numerous times with various right hand sides f . We proposehere a general framework for rapidly constructing and evaluating good ABCs bycompressing the Dirichlet to Neumann (DtN) map D,
D : u(s)→ ∂u
∂ν(x)
where u is the desired solution, ν the outward pointing normal to the boundaryof interest Γ, and s, x ∈ Γ. This is based on our previous work on the DtN mapfor the Helmholtz equation, which showed that this map is separable and low-rankfor the constant media half-space case when x and s are sufficiently well separated.Once we have obtained a compressed and accurate approximation D to the DtNmap D, it can repeatedly be used in any Helmholtz solver for a bounded domain inorder to simulate an unbounded domain. Namely, one uses the following boundarycondition
∂u
∂ν(s)− Du(s) = 0, s ∈ Γ
on any boundary Γ for which one wishes to have an absorbing boundary condi-tion. We use matrix probing in order to rapidly solve for the DtN map, and thisconveniently gives us a compressed map. The computational complexity of thisstep is equivalent to solving the original problem, with the desired ABC and a zeroright hand side, a few times only. Since matrix probing fits the DtN map to a fewwell-chosen matrices, it has greater potential for accuracy and flexibility in variablemedia. Hence we may use matrix probing on the DtN map as a precomputationfor solving the wave equation multiple times on unbounded domains in variablemedia.
115
Curl-Conforming Hardy Space Infinite Elements for ExteriorMaxwell Problems
Session Schedule Author Index Session Index M9
Achim SchadleHeinrich-Heine Universitat [email protected]
Thorsten HohageGeorg-August Universitat [email protected]
Lothar NannenTechnische Universitat [email protected]
Joachim SchoberlTechnische Universitat [email protected]
AbstractThe construction of a new kind of prismatic infinite elements for electromagneticscattering and resonance problems will be described. Transparent boundary condi-tions are realized by the pole condition. The pole condition, as radiation condition,states that a function is outgoing if and only if a certain transformation of thisfunction belongs to a Hardy space. We use tensor products of cochain complexesto obtain four different infinite element spaces which form an exact sequence corre-sponding to the deRham complex in the exterior domain. Numerical tests indicatesuper-algebraic convergence in the number of additional unknows per degree offreedom on the coupling boundary.
116
Finite Element Construction of Discrete Harmonic Fields
Session Schedule Author Index Session Index M9
Ana Alonso RodriguezDepartment of Mathematics - University of [email protected]
Enrico BertolazziDepartment of Mechanical and Structural Engineering - University of [email protected]
Riccardo GhiloniDepartment of Mathematics - University of [email protected]
Alberto ValliDepartment of Mathematics - University of [email protected]
AbstractWe study an efficient algorithm for the finite element construction of discrete har-monic fields in a not simply-connected bounded three-dimensional domain Ω. Theconstruction of discrete harmonic fields is essential in the numerical approximationof magnetostatic and H-based formulations of the eddy-current model. It is wellknown that the dimension of the space of harmonic fields is equal to the first Bettinumber of Ω, nΩ. The fundamental point in the classical way for determining abasis of the space of harmonic fields is the fact that there exists nΩ connected ori-entable Lipschitz surfaces Σn, with ∂Σn ⊂ ∂Ω, such that every curl-free vector fieldin Ω has a global potential in Ω \ ∪nΣn. These surfaces are called Seifert surfaces:each one of them “cuts” a non-bounding cycle in Ω. From the computational pointof view, in general topological situations it can be difficult to explicitly determinethe Seifert surfaces (for instance, in the case of “knotted” domains). We propose analternative method that works for general topological domains and does not needthe determination of cutting surfaces.
117
Hybrid Domain Decomposition Solvers for the HelmholtzEquation
Session Schedule Author Index Session Index M9
Martin HuberInstitute for Analysis and Scientific Computing,Vienna University of [email protected]
Joachim SchoberlInstitute for Analysis and Scientific Computing,Vienna University of [email protected]
AbstractIn this talk, an hybrid finite element methods (FEM) for the scalar and vectorialwave equation, which is equivalent to a discontinuous Gelerkin Method, based onthe Ultra Weak Variational Formulation is investigated.Motivated by hybrid FEMs for the Laplace equation, the tangential continuity ofthe flux is broken across element interfaces. In order to reinforce continuity again,Lagrange multipliers supported only on the element facets are introduced. Thesemultipliers can be interpreted as the tangential component of the unknown field.By adding a second set of multipliers, representing the tangential component of theflux field, it is possible to eliminate the volume degrees of freedom cheaply elementby element. This approach allows to reduce the original system of equations to amuch smaller system for the Lagrange multipliers.A very challenging point is to solve the resulting system of equations. Differentpreconditioners for Krylov space solvers are discussed in the talk. Apart frommultiplicative and additive Schwarz block preconditioners with blocks containingdegrees of freedom related to one facet or element, respectively, or an elementwiseBDDC preconditioner, a domain decomposition preconditioner is constructed. Thispreconditioner solves in each iteration step local problems on subdomains by di-rectly inverting the system matrix, and it can be therefore implemented efficientlyin parallel.It is shown by numerical experiments, that the iterative solvers have good conver-gence properties for large scale computations. Our approach allows the solution ofthree dimensional problems up to 50 wavelength per domain for the scalar Helmholtzequation, and up to 30 wavelength per domain for the vectorial wave equation.
118
M9P4 Schedule
Fast Solvers for Helmholtz and Maxwell equations
Schedule Author Index Session Index
Date: Wednesday, June 27Time: 16:00-17:45Location: TuringChairman: Victorita Dolean, Ronan Perrussel, Hui Zhang, Peng Zhen
16:05-16:30 : Ronan PerrusselSchwarz Methods for Time-Harmonic Maxwell’s Equations Discretized by aHybridized Discontinuous Galerkin MethodAbstract
16:30-16:55 : Jack PoulsonA Parallel Sweeping Preconditioner for High-Frequency Heterogeneous 3dHelmholtz EquationsAbstract
16:55-17:20 : Zhen PengSpeed up Non-conformal DDM Convergence using an Asymmetric OptimalTransmission ConditionAbstract
17:20-17:45 : Stephane LanteriDiscretization of Optimized Schwarz Methods for Maxwell’s EquationsAbstract
M9 Abstract
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119
Schwarz Methods for Time-Harmonic Maxwell’s EquationsDiscretized by a Hybridized Discontinuous Galerkin Method
Session Schedule Author Index Session Index M9
Ronan PerrusselCNRS/Universite de [email protected]
Stephane [email protected]
Liang LiUniversity of Electronic Science and Technology of China, [email protected]
AbstractSchwarz-type domain decomposition methods are presented for the solution of 3dtime-harmonic Maxwell’s equations. They are coupled with a hybridizable dis-continuous Galerkin (HDG) method for the discretization of the problem. Theformulation and the implementation of the HDG method associated with Schwarzalgorithms are detailed. Numerical results show that the HDG method has an op-timal convergence rate and can save both CPU time and memory cost compared tomore classical DG methods.
120
A Parallel Sweeping Preconditioner for High-FrequencyHeterogeneous 3d Helmholtz Equations
Session Schedule Author Index Session Index M9
Jack PoulsonUniversity of Texas at [email protected]
Lexing YingUniversity of Texas at [email protected]
Bjorn EngquistUniversity of Texas at [email protected]
Sergey FomelUniversity of Texas at [email protected]
Siwei LiUniversity of Texas at [email protected]
AbstractA parallelization of a recently introduced ‘sweeping’ preconditioner for high-frequencyheterogeneous Helmholtz equations is presented along with experimental results forthe full SEG/EAGE Overthrust seismic model at 30 Hz. While the setup and appli-cation costs of the sweeping preconditioner are trivially O(N4/3) and O(N logN),this study provides strong empirical evidence for Engquist and Ying’s observationthat the number of iterations required for the convergence of GMRES with thesweeping preconditioner is essentially independent of the frequency of the prob-lem. Generalizations to more complicated time-harmonic wave equations are alsobriefly discussed since the techniques behind our parallelization are not specific tothe Helmholtz equation.
121
Speed up Non-conformal DDM Convergence using anAsymmetric Optimal Transmission Condition
Session Schedule Author Index Session Index M9
Zhen PengElectroScience Laboratory, The Ohio State [email protected]
Jin-Fa LeeElectroScience Laboratory, The Ohio State [email protected]
Victorita DoleanSection de mathematiques, Universite de [email protected]
Martin J. GanderSection de mathematiques, Universite de [email protected]
Stephane LanteriNACHOS project-team, INRIA Sophia [email protected]
AbstractIt has been recognized that the convergence of non-overlapping DDMs dependsheavily upon the transmission conditions employed to enforce the continuity oftangential fields on the domain interfaces. Recently, a second order transmissioncondition and an optimal transmission condition, which involve two second-ordertransverse derivatives, are proposed to enable convergence for both propagating andevanescent electromagnetic waves across domain interfaces. Additionally, by solv-ing the second-order vector wave equation, we have placed more emphasis on onevector field than the other. For example, in the so-called E-field formulation, whichis the formulation adopted herein, we have observed the accuracy in the electricfield computed is better than the accuracy obtained in the magnetic field. Thisdiscrepancy in accuracy leads to different effective frequency spectrum for both Eand H fields for finite discretization. To exploit this observation, we further adjustthe parameters in the optimal transmission condition to result in an asymmetrictransmission condition which has different spectral convergences for both trans-verse electric (TE) and transverse magnetic (TM) fields, respectively. Furthermore,through an analysis of the DDM with the higher order TCs, we show that there stillexists a weakly converging region centered on the cutoff modes at each sub-domaininterfaces. A global plane wave deflation technique is then proposed as an effectivecoarse grid preconditioner.
122
Discretization of Optimized Schwarz Methods for Maxwell’sEquations
Session Schedule Author Index Session Index M9
Stephane [email protected]
Victorita DoleanUniversite de Nice and Universite de [email protected]
Martin J. GanderUniversite de [email protected]
Ronan PerrusselCNRS, Universite de [email protected]
AbstractWe study here optimized Schwarz domain decomposition methods for solving thetime-harmonic Maxwell equations discretized by a discontinuous Galerkin (DG)method. Due to the particularity of the latter, a discretization of a more so-phisticated Schwarz method is not straightforward. A strategy of discretizationis shown in the framework of a DG weak formulation, and the equivalence be-tween multi-domain and single-domain solutions is proved. The proposed discreteframework is then illustrated by some numerical results through the simulation oftwo-dimensional propagation problems in homogeneous and heterogeneous media.
123
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Mini Symposium M10New Developments of FETI, BDDC, and Re-lated Domain Decomposition MethodsSchedule Author Index Session Index
Organizers: Xuemin Tu and Olof Widlund
AbstractFinite Element Tearing and Interconnecting (FETI) and Balancing Domain Decom-position by Constraints (BDDC) methods are among the most powerful domaindecomposition methods. They are also closely related in that they use the same orsimilar coarse components in their preconditioners. Lately, coarse solvers borrowedfrom these and other iterative substructuring algorithms have also been combinedwith local solvers defined on a set of overlapping subdomains. In this mini sympo-sium, some of this new work will be discussed. Among the applications are Stokes’equations, elasticity, some problems posed in H(curl), discontinuous Galerkin meth-ods, and isogeometric analysis.
Part 1
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M10P1 Schedule
New Developments of FETI, BDDC, and Related Do-main Decomposition Methods
Schedule Author Index Session Index
Date: Monday, June 25Time: 10:30-12:15Location: MarkovChairman: Xuemin Tu and Olof Widlund
10:35-11:00 : Maksymilian DryjaASM for DG Discretization of Anisotropic Elliptic ProblemsAbstract
11:00-11:25 : Juan GalvisDomain Decomposition Preconditioners for High-Contrast MultiscaleProblemsAbstract
11:25-11:50 : Hyea Hyun KimTwo-Level Overlapping Schwarz Algorithms for a Staggered DiscontinuousGalerkin MethodAbstract
11:50-12:15 : Chang-Ock LeeA Two-Level Nonoverlapping Schwarz Algorithm for the Stokes ProblemAbstract
M10 Abstract
Part 2
126
ASMs for DG Discretization of Anisotropic Elliptic Problems
Session Schedule Author Index Session Index M10
Max DryjaUniversity of [email protected]
Marcus SarkisWorcester Polytechnic [email protected]
Piotr KrzyzanowskiUniversity of [email protected]
AbstractIn the talk a Discontinuous Galerkin (DG) discretization of an elliptic second orderequation with discontinuous anisotropic coefficients is considered. The DG dis-cretization is applied elementwise, on a matching triangulation.First, an analysis of the discrete problem will be presented, including the errorbound. Next, Additive Schwarz Methods (ASMs) will be designed and analyzed,with rates of the convergence independent of the jumps of the coefficients.The preliminary ASM for the discussed problem is determined by local problemsdefined on each triangle and a coarse problem defined in the space of piecewise con-stant functions on the fine mesh. This preconditioner is optimal and independentof te jumps of the coefficients.To get a fully parallel algorithm, the coarse problem of the above method can besolved by known ASMs. This leads to various new ASMs. One of them, whichconvergence rate is independent of the coefficient jumps is discussed in detail.
127
Domain Decomposition Preconditioners for High-ContrastMultiscale Problems
Session Schedule Author Index Session Index M10
Juan GalvisTexas A&M [email protected]
Yalchin EfendievTexas A&M [email protected]
AbstractWe give an overview of our results from the point of view of coarse-grid multiscalemodel reduction by highlighting some common issues in coarse-scale approximationsand two-level preconditioners. Reduced models discussed in this paper rely oncoarse-grid spaces computed by solving local spectral problems. We define localspectral problems with a weight function computed with a choice of initial multiscalebasis functions. We emphasize the importance of this initial choice of multiscalebasis functions for both coarse-scale approximation and for preconditioners. Inparticular, we discuss various choices of initial basis functions and use some of themin our simulations. We show that a naive choice of initial basis functions, e.g.,piecewise linear functions, can lead to a large dimensional spaces that are needed toachieve (1) a reasonable accuracy in the coarse-scale approximation or (2) contrast-independent condition number of preconditioned matrix within two-level additiveSchwarz methods. While using a careful choice of initial spaces, we can achieve (1)and (2) with smaller dimensional coarse spaces.
128
Two-Level Overlapping Schwarz Algorithms for a StaggeredDiscontinuous Galerkin Method
Session Schedule Author Index Session Index M10
Hyea Hyun KimDepartment of Applied Mathematics, Kyung Hee University, [email protected]
Eric T. ChungDepartment of Mathematics, The Chinese University of Hong [email protected]
Olof B. WidlundCourant Institute of Mathematical Sciences, [email protected]
AbstractTwo overlapping Schwarz algorithms are developed for a discontinuous Galerkin(DG) finite element approximation of second order scalar elliptic problems in bothtwo and three dimensions. The discontinuous Galerkin formulation is based on astaggered discretization introduced by Chung and Engquist for the acoustic waveequation. Two types of coarse problems are introduced for the two-level Schwarzalgorithms. The first is built on a nonoverlapping subdomain partition, which allowsquite general subdomain partitions, and the second on introducing an additionalcoarse triangulation that can also be quite independent of the fine triangulation.Condition number bounds are established and numerical results are presented.
129
A Two-Level Nonoverlapping Schwarz Algorithm for the StokesProblem
Session Schedule Author Index Session Index M10
Chang-Ock [email protected]
Hyea Hyun KimKyung Hee [email protected]
AbstractA general framework of a two-level nonoverlapping Schwarz algorithm for the Stokesproblem is developed. This framework allows both discontinuous and continuouspressure finite element spaces. The coarse problem is built by algebraic manipu-lation after selecting appropriate primal unknowns just like in BDDC algorithms.Performance of the suggested algorithm is presented depending on the selection offinite elements and primal unknowns. Under the same set of primal unknowns, thealgorithm for the case with discontinuous pressure functions outperforms one withcontinuous pressure functions. For the two-dimensional Stokes problem, the algo-rithm with a set of primal unknowns consisting of velocity unknowns at corners,averages of velocity components over common edges, and pressure unknowns atcorners presents good scalability when continuous pressure test functions are used.In both two- and three-dimensional Stokes problems, an improvement can be madefor the case with continuous pressure test functions by applying the suggested algo-rithm to the interface problem, which is obtained by eliminating velocity unknownsand pressure unknowns interior to each subdomains.
130
M10P2 Schedule
New Developments of FETI, BDDC, and Related Do-main Decomposition Methods
Schedule Author Index Session Index
Date: Monday, June 25Time: 16:00-18:10Location: MarkovChairman: Xuemin Tu and Olof Widlund
16:05-16:30 : Jungho LeeLarge-Scale Differential Variational Inequalities for Phase-Field ModelingAbstract
16:30-16:55 : Lourenco Beirao da VeigaBDDC Preconditioners for Isogeometric AnalysisAbstract
16:55-17:20 : Xuemin TuFETI-DP Domain Decomposition Methods for Incompressible StokesEquationAbstract
17:20-17:45 : Olof WidlundBDDC for some problems posed in H(curl)Abstract
17:45-18:10 : Jun ZouAn Overlapping Domain Decomposition Algorithm for ParameterIdentificationsAbstract
M10 Abstract
Part 1
131
Large-Scale Differential Variational Inequalities for Phase-FieldModeling
Session Schedule Author Index Session Index M10
Jungho LeeArgonne National [email protected]
Shrirang AbhyankarArgonne National [email protected]
Mihai AnitescuArgonne National [email protected]
Lois McInnesArgonne National [email protected]
Todd MunsonArgonne National [email protected]
Barry SmithArgonne National [email protected]
Lei WangArgonne National [email protected]
AbstractRecent progress on the development of scalable differential variational inequalitymultigrid-based solvers for the phase-field approach to mesoscale materials modelingis described. We have developed an active-set method for variational inequalities inPETSc, leveraging experience by the optimization community in TAO. A geometricmultigrid solver in PETSc is used to solve the resulting linear systems. We presentstrong and weak scaling results for 2D coupled Allen-Cahn/Cahn-Hilliard systems.
132
BDDC Preconditioners for Isogeometric Analysis
Session Schedule Author Index Session Index M10
Lourenco Beirao da VeigaUniversita di Milano, [email protected]
Durkbin ChoDongguk University, Seoul, South [email protected]
Luca Franco PavarinoUniversita di Milano, [email protected]
Simone ScacchiUniversita di Milano, [email protected]
AbstractWe study Balancing Domain Decomposition by Constraints (BDDC) precondition-ers for NURBS-based Isogeometric Analysis of scalar elliptic and linear elasticityproblems. Our construction is based on a generalized Schur complement systemfor NURBS functions and the analysis is based on appropriate discrete norms andscaling functions. The proposed isogeometric BDDC preconditioner with differentchoices of primal constraints is proven to be scalable in the number of subdomainsand quasi-optimal in the ratio of subdomain and element sizes. Several numeri-cal experiments in 2D and 3D confirm the theoretical convergence rate estimatesobtained and also illustrate the preconditioner performance with respect to thepolynomial degree and regularity of the NURBS basis functions, as well as its ro-bustness with respect to discontinuities of the elliptic coefficients across subdomainboundaries.
References:
L. Beirao da Veiga, D. Cho, L.F. Pavarino, S. Scacchi, BDDC preconditioners forIsogeometric Analysis. Technical report IMATI-CNR, 2012
133
FETI-DP Domain Decomposition Methods for IncompressibleStokes Equation
Session Schedule Author Index Session Index M10
Xuemin TuUniversity of [email protected]
Jing LiKent State [email protected]
AbstractIn this talk, a unified framework of FETI-DP algorithms will be discussed for solvingthe system of linear equations arising from the mixed finite element approximation ofincompressible Stokes equations. Several previously developed FETI-DP algorithmscan be represented under this framework. Their condition number estimates arealso simplified using this framework. A distinctive feature of this framework is thatboth continuous and discontinuous pressures can be used in the finite element space,while previous FETI-DP algorithms are valid only for the case of using discontinuouspressures. Both lumped and Dirichlet type preconditioners will be discussed andnumerical experiments of solving a two-dimensional incompressible Stokes problemwill be provided.
134
BDDC for some problems posed in H(curl)
Session Schedule Author Index Session Index M10
Olof B. WidlundCourant [email protected]
Clark DohrmannSandia National [email protected]
AbstractThe considerable challenge in developing domain decomposition algorithms for prob-lems posed in H(curl) and discretized using Nedelec edge elements is reflected ina quite sparse literature. This talk will discuss some of these challenges and newresults recently developed for BDDC algorithms. These results improve and gener-alize findings by Toselli on the closely related FETI–DP algorithms and they alsosimplify some of his arguments.
135
An Overlapping Domain Decomposition Algorithm forParameter Identifications
Session Schedule Author Index Session Index M10
Jun ZouThe Chinese University of Hong [email protected]
Hui FengWuhan [email protected]
Daijun JiangWuhan [email protected]
AbstractIn this talk we shall address an overlapping domain decomposition method forsolving some inverse problems of identifying parameters in second order ellipticand parabolic systems, including the reconstruction of fluxes, heat sources andinitial data. Some convergence analysis and numerical experiments of the proposedmethod will be presented and discussed. The work of Jun Zou was substantiallysupported by Hong Kong RGC grants (Projects 405110 and 404611)
136
Mini Symposium M11Decomposition Strategies for Boltzmann’s Equa-tionSchedule Author Index Session Index
Organizers: Heiko Berninger and Jerome Michaud
AbstractSolving Boltzmann’s equation for non-uniform gases or for neutrino transport is achallenging task if large variations of the mean free path have to be considered. Inparticular, computation times become excessively large in high density regimes, i.e.,if the mean free path is small. As a remedy, one uses suitable macroscopic limitequations such as Euler, Navier–Stokes or diffusion equations in regimes wheremicroscopic kinetic effects may be neglected or considered in average. Then thequestion arises of how to couple these equations and Boltzmann’s equation prop-erly and how to solve the coupled system.
With regard to this problem of solving Boltzmann’s equation in its different regimes,various decomposition and domain decomposition strategies have been establishedand investigated. This minisymposium intends to bring experts together who rep-resent a variety of different approaches. Our aim is to obtain an overview on theachievements in this field and to discuss main challenges and open problems.
Part 1
Part 2
137
M11P1 Schedule
Decomposition Strategies for Boltzmann’s Equation
Schedule Author Index Session Index
Date: Tuesday, June 26Time: 10:30-12:15Location: I50Chairman: Heiko Berninger and Jerome Michaud
10:35-11:00 : Patrick Le TallecHalf Fluxes Coupling of Boltzmann and Navier Stokes EquationsAbstract
11:00-11:25 : Mohammed LemouOn Micro-Macro Numerical Schemes for Multiscale Kinetic EquationsAbstract
11:25-11:50 : Emmanuel FrenodTwo-Scale Convergence and Kinetic EquationsAbstract
11:50-12:15 : Heiko BerningerNeutrino Transport in Core Collapse Supernovae by Asymptotic Expansionsof Boltzmann’s EquationAbstract
M11 Abstract
Part 2
138
Half Fluxes Coupling of Boltzmann and Navier Stokes Equations
Session Schedule Author Index Session Index M11
Patrick Le TallecEcole [email protected]
AbstractMany practical problems require the simultaneous use of different physical modelsinside a given computational domain:- local kinetic models must be used in shock or boundary layers when simulatingrarefied flows or radiative effects,- refined meshes or boundary layer models are needed in recirculation regions ornext to solid boundaries.Such situations are nicely treated by a general purpose domain decomposition strat-egy largely developed and applied by Quarteroni and coauthors, and based on thefollowing steps:- introduction of a unique underlying system of differential equations modelling allcomponents of the system under study ;- consistent multidomain splitting of the underlying Partial Differential Equations.The coupling conditions are deduced from the mathematical structure of the under-lying mathematical model. The subdomains can be either constructed apriori bygeometric or physical arguments, or adaptively updated during the solution process;- introduction of local approximation strategies independently on each subdomain.The proposed talk will review the application of this general strategy to kineticmodels coupled to hydrodynamics simulations. The underlying physical model willbe based on a full Boltzmann eaquation or on a fourteen moment asymptotic ex-pansion of the Boltzmann equation proposed by D. Levermore. The Navier-Stokesequations can then be obtained by a Hilbert asymptotic expansion of these kineticequations. Following the above general strategy lead then to coupling strategieswhich acts as if we were solving a kinetic model everywhere, the Navier-Stokesequations being simply obtained locally by replacing the kinetic model by theirspecific forms derived by the appropriate asymptotic expansion process.
139
On Micro-Macro Numerical Schemes for Multiscale KineticEquations
Session Schedule Author Index Session Index M11
Mohammed LemouCNRS, IRMAR, University of Rennes [email protected]
AbstractThe development of numerical methods to solve multi-scale kinetic equations hasbeen the subject of active research in the past years, with applications in variousfields: plasma physics, rarefied gas dynamics, aerospace engineering, semiconduc-tors, radiative transfer, ... The general problem is to construct numerical schemesthat are able to capture the properties of the various scales in the considered system,while the numerical parameters remain as independent as possible of the stiffnesscharacter of these scales. In systems of particles, on could have to deal for instancewith various regimes in different regions of the physical space: Microscopic regime(kinetic) or macroscopic regimes (fluid, diffusion,etc...). In contrast with a domaindecomposition method (with respect to these different regimes), which would re-quire a delicate handling of the interfaces, we have developed (in 2008) a numericalmethod which is based on a decomposition of the distribution function in the wholespace. The original model is then decomposed into a system of two equations:an equation on a macro part (equilibrium part) whose evolution is coupled to anequation on the remaining micro (kinetic) part. Suitable numerical schemes arethen constructed on this formulation. This strategy is quite robust in the sensethat it can be easily adapted to a large class of kinetic equations (Vlasov-BGK,Vlasov-Boltzmann, Vlasov-Landau, etc) and to various scales (kinetic/diffusion, ki-netic/fluid kinetic/high fields, two-scales asymptotics, etc). The numerical schemeswhich are constructed from such formulations have the following so-called asymp-totic preserving (AP) property: they are consistent with the model at the kineticregime and they degenerate into a scheme which is consistent with the macro modelin the desired asymptotic limit, the numerical parameters being fixed. In this talk,we shall first present the main lines of this strategy for different asymptotics. Then,we will show how to modify the approach in order to deal with the space boundaryconditions and to provide good approximations of boundary layers as well. Thislast point was the subject of a recent work on which a main part of the talk will befocused.
140
Two-Scale Convergence and Kinetic Equations
Session Schedule Author Index Session Index M11
Emmanuel FrenodUniversite de [email protected]
AbstractIn this talk, we will explain how Numerical Methods mixing Two-Scale Convergenceand Macro-Micro decomposition can be set out for Kinetic Equation with a singularterm generating large amplitude and high frequency oscillations in some regions ofthe domain where the Equation is defined, but not everywhere.The presented work follows [3] where a Two-Scale Numerical method was built forthe 2D-Vlasov-Poisson system. This method is well adapted to Kinetic Equationwith a singular term which is uniform over the domain.In [1] we proposed a strategy to develop Two-Scale Macro-Micro decompositionthat is a path to the desired Numerical Methods. A simplified version or the desiredNumerical Methods was built in [2].
[1] N. Crouseilles, E. Frenod, S. Hirstoaga, and A. Mouton. Two-Scale Macro-Microdecomposition of the Vlasov equation with a strong magnetic field. Submitted.
[2] E. Frenod, M. Gutnic, and S. Hirstoaga. First order Two-Scale Particle-in-Cell numerical method for Vlasov equation. Submitted to: ESAIM Proceedings ofCEMRACS 2011, January 2012.
[3] E. Frenod, F. Salvarani, and E. Sonnendrucker. Long time simulation of a beamin a periodic focusing channel via a two-scale PIC-method. Mathematical Modelsand Methods in Applied Sciences, 19(2):175–197, 2009.
141
Neutrino Transport in Core Collapse Supernovae by AsymptoticExpansions of Boltzmann’s Equation
Session Schedule Author Index Session Index M11
Heiko BerningerUniversite de [email protected]
Emmanuel FrenodUniversite de [email protected]
Martin J. GanderUniversite de [email protected]
Jerome MichaudUniversite de [email protected]
AbstractSimulations of core collapse supernovae require to model the radiative transfer ofneutrinos. If the full Boltzmann equation is chosen as the model, computationtimes increase dramatically in regimes in which high density of neutrinos, i.e., smallmean free paths prevail. In core collapse supernovae, neutrinos are trapped bythe matter in high density regimes, where they are in diffusion, whereas they arepractically freely streaming in low density regimes further away from the core. Inboth cases, the Boltzmann equation can be reduced considerably. This leads tothe idea of a decomposition of the neutrino distribution function into trapped andstreaming particle components used in the Isotropic Diffusion Source Approximation(IDSA) of Boltzmann’s equation [?]. In this talk, we will give an introductioninto the IDSA in spherical symmetry, both from a physical and a mathematicalpoint of view. The main purpose of the talk will be to present a derivation of theIDSA by asymptotic analysis applying Chapman–Enskog and Hilbert expansions.Computational aspects and numerical solution techniques for the IDSA and theBoltzmann equation accompanied by numerical results will be subject of the talkby J. Michaud in the same minisymposium.
[1] M. Liebendorfer, S.C. Whitehouse, and T. Fischer. The Isotropic DiffusionSource Approximation for Supernova Neutrino Transport. ApJ, 698:1174–1190,2009.
142
M11P2 Schedule
Decomposition Strategies for Boltzmann’s Equation
Schedule Author Index Session Index
Date: Tuesday, June 26Time: 16:00-17:45Location: I50Chairman: Heiko Berninger and Jerome Michaud
16:05-16:30 : Francois GolseA Coupling Method for Transport/Diffusion ProblemsAbstract
16:30-16:55 : Giacomo DimarcoFluid Simulations with Localized Boltzmann Upscaling by Direct MonteCarloAbstract
16:55-17:20 : Sudarshan TiwariSimulation of the Boltzmann and the Navier-Stokes Equations with ParticleMethods based on Domain Decomposition for Steady and Unsteady FlowsAbstract
17:20-17:45 : Jerome MichaudThe IDSA and Boltzmann’s Equation: Discretization, Comparison andModeling ErrorAbstract
M11 Abstract
Part 1
143
A Coupling Method for Transport/Diffusion Problems
Session Schedule Author Index Session Index M11
Francois GolseEcole polytechnique, Centre de Mathematiques L. Schwartz, 91128 Palaiseau, [email protected]
AbstractThis talk presents a domain decomposition method on an interface problem for thelinear transport equation between a diffusive and a non-diffusive region. To leadingorder, i.e. up to an error of the order of the particle mean free path in the diffusiveregion, the solution in the non-diffusive region is decoupled from the particle densityin the diffusive region. The diffusive and non-diffusive regions are coupled at theinterface at the next order of approximation in the mean-free path. Our analysis isbased on an accurate description of the boundary layer at the interface, in termsof a half-space problem for the linear transport equation. Indeed, such half-spaceproblems for the linear transport equation can be reduced to a linear integral equa-tion of Wiener-Hopf type and can be solved explicitly. We take advantage of a verysimple formulation of this solution due to Chandrasekhar.
144
Fluid Simulations with Localized Boltzmann Upscaling by DirectMonte Carlo
Session Schedule Author Index Session Index M11
Giacomo DimarcoUniversite de Toulouse; UPS, INSA, UT1, UTM; CNRS, UMR 5219; Institut deMathematiques de Toulouse; F-31062 Toulouse, [email protected]
Pierre DegondUniversite de Toulouse; UPS, INSA, UT1, UTM; CNRS, UMR 5219; Institut deMathematiques de Toulouse; F-31062 Toulouse, [email protected]
AbstractIn the present talk, we present a novel numerical algorithm to couple the DirectSimulation Monte Carlo method (DSMC) for the solution of the Boltzmann equationwith a finite volume like method for the solution of the Euler equations. The methodrelies on the introduction of buffer zones which realize a smooth transition betweenthe kinetic and the fluid regions. To facilitate the coupling and avoid the onset ofspurious oscillations in the fluid regions which are consequences of the coupling witha stochastic numerical scheme, we use a new technique which permits to reduce thevariance of the particle methods. In addition, the use of this method permits toobtain estimations of the breakdowns of the fluid models less affected by fluctuationsand consequently to reduce the kinetic regions and optimize the coupling. In thelast part of the talk several numerical examples are presented to validate the methodand measure its computational performances.
145
Simulation of the Boltzmann and the Navier-Stokes Equationswith Particle Methods based on Domain Decomposition forSteady and Unsteady Flows
Session Schedule Author Index Session Index M11
Sudarshan TiwariDepartment of Mathematics, TU Kaiserslautern, [email protected]
Axel KlarDepartment of Mathematics, TU Kaiserslautern, [email protected]
Steffen HardtCenter of Smart Interfaces, TU Darmstadt , [email protected]
AbstractWe present a coupling of the Boltzmann and the Navier-Stokes equations based onthe domain decomposition strategy. We monitor the breakdown criterion of thecontinuum regime in time and decompose the domains. The Boltzmann equationis solved in the rarefied domain and the Euler/Navier-Stokes equations are solvedin the continuum domain. We use a DSMC type of particle method to solve theBoltzmann equation. Earlier we have solved the compressible Euler equations by akinetic particle method in the continuum regime. This method is a natural choice forcoupling since the two schemes defer only in the collision processes and it is easierto handle the interface boundary conditions. However, kinetic particle methodsare not the optimal choice. In recent years we have extended our earlier work insmall scale geometries for unsteady as well as steady problems. In the continuumregime we solve the Navier-Stokes equations by a meshfree particle method. In thisscenario meshfree methods are suitable since the interface between the continuumand rarefied domains is quite irregular. Meshfree methods are capable of handlingsuch irregular boundaries easily. The classical Sod’s shock tube problem is solvedas a 1D test case, whereas as 2D test case a stationary driven cavity flow is solvedfor a large range of Knudsen numbers. For small Knudsen numbers all solutionsobtained from pure Boltzmann, pure Navier-Stokes and the couplied solvers matchperfectly. We further show that for larger Knudsen numbers, where the Navier-Stokes equations fail to predict the correct flow behavior, its solutions are still agood candidate to initialize a Boltzmann solver. Finally, we show the simulation ofa moving droplet inside a rarefied regime.
146
The IDSA and Boltzmann’s Equation: Discretization,Comparison and Modeling Error
Session Schedule Author Index Session Index M11
Jerome MichaudUniversite de [email protected]
Heiko BerningerUniversite de [email protected]
Emmanuel FrenodUniversite de [email protected]
Martin J. GanderUniversite de [email protected]
AbstractIn this talk we recall the Boltzmann equation for neutrino transport used in core col-lapse supernovae as well as the Isotropic Diffusion Source Approximation (IDSA) ofit [1]. The latter is presented and analysed in more detail in the talk of H. Berningerin this minisymposium. The purpose of this talk is to present a numerical treatmentof a reduced Boltzmann model problem based on time splitting and finite volumesand revise the discretization of the IDSA for this problem [2]. Discretization errorstudies carried out on the reduced Boltzmann model problem and on the IDSAreveal errors of order one in both cases. By means of a numerical example, a de-tailed comparison of the reduced model and the IDSA is performed and interpreted.For this example, the IDSA modeling error with respect to the reduced Boltzmannmodel is numerically determined and localized.
[1] M. Liebendorfer, S.C. Whitehouse, and T. Fischer. The Isotropic DiffusionSource Approximation for Supernova Neutrino Transport. ApJ, 698:1174–1190,2009.
[2] H. Berninger, E. Frenod, M.J. Gander, M. Liebendorfer, J. Michaud, and N. Vas-set. A Mathematical Description of the IDSA for Supernova Neutrino Transport,its Discretization and a Comparison with a Finite Volume Scheme for Boltzmann’sEquation. Submitted to: ESAIM Proceedings of CEMRACS 2011.
147
148
Mini Symposium M12Domain Decomposition Techniques in Life Sci-ence Modeling and SimulationSchedule Author Index Session Index
Organizers: Luca Gerardo-Giorda and Victorita Dolean
AbstractIn the recent decades, the availability of powerful computers led scientists to faceproblems coming from biomedical applications, and computational science has be-come a strong partner for medical doctors. Biomedical problems are very challeng-ing in terms of both computational and mathematical complexity. Advanced geom-etry reconstruction techniques from medical imaging made very detailed anatomiesavailable. If on the one hand in silico experiments based on real domains havebecome quite a standard in the community, on the other hand such a detaileddescription of the computational domain easily results in millions of degrees of free-dom. At the same time, most problems in Life Science modeling aim at describingthe coupling of different physiological and mechanical models. In both cases, do-main decomposition methods are the most natural environment to both formulateand solve such problems. The minisymposium aims at gathering researchers whobrought important scientific contributions in this field.
149
M12 Schedule
Domain Decomposition Techniques in Life Science Mod-eling and Simulation
Schedule Author Index Session Index
Date: Friday, June 29Time: 10:30-12:15Location: I50Chairman: Luca Gerardo-Giorda and Victorita Dolean
10:35-11:00 : Oliver RheinbachAdvances of FETI Methods in BiomechanicsAbstract
11:00-11:25 : Simone ScacchiParallel Bidomain Solvers for Cardiac ExcitationAbstract
11:25-11:50 : Nejib ZemzemiDecoupled Time-Marching Schemes in Computational CardiacElectrophysiology and ECG Numerical SimulationAbstract
11:50-12:15 : Gwenol GrandperrinParallel Preconditioners for Solving Fluid-Structure Interactions Problems inHemodynamicsAbstract
150
Advances of FETI Methods in Biomechanics
Session Schedule Author Index Session Index M12
Oliver RheinbachFakultat fur Mathematik, TU [email protected]
Sarah BrinkhuesInstitut fur Mechanik, Universitat [email protected]
Andreas FischleFakultat fur Mathematik, Universitat [email protected]
Axel KlawonnMathematisches Institut, Universitat zu [email protected]
Jorg SchroderInstitut fur Mechanik, Universitat [email protected]
AbstractIn this talk we will present recent results on parallel simulations of soft biologi-cal tissue obtained by using a FETI-DP method within a fully parallel softwareenvironment built around FEAP. We will present new parallel scalability resultson a Cray XT6m and will investigate well known alternatives to the load steppingschemes traditionally used in nonlinear structural mechanics within this context.
151
Parallel Bidomain Solvers for Cardiac Excitation
Session Schedule Author Index Session Index M12
Simone ScacchiUniversity of [email protected]
Piero Colli FranzoneUniversity of [email protected]
Luca Franco PavarinoUniversity of [email protected]
AbstractThe Bidomain model of electrocardiology describes the bioelectric activity of thecardiac muscle and consists of a parabolic degenerate system of non-linear partialdifferential equation (PDE). The PDEs are coupled with a system of ordinary dif-ferential equations (ODEs), modeling the cellular membrane ionic currents. Thediscretization of the Bidomain model in three-dimensional (3D) cardiac geometriesyields the solution of large scale and ill-conditioned linear systems at each timestep. The aim of this work is to develop parallel multilevel and block precondi-tioners, in order to reduce the high computational costs required by the solutionof the Bidomain model in 3D domains of realistic size. We analyze the scalabilityof multilevel Schwarz block-diagonal and block-factorized preconditioners for theBidomain model and compare them with multilevel Schwarz coupled precondition-ers. 3D parallel numerical tests show that block preconditioners are scalable, butless efficient than the coupled preconditioners.
152
Decoupled Time-Marching Schemes in Computational CardiacElectrophysiology and ECG Numerical Simulation
Session Schedule Author Index Session Index M12
Nejib ZemzemiInria Bordeaux [email protected]
Miguel FernandezInria [email protected]
AbstractThis work considers the approximation of the cardiac bidomain equations, either iso-lated or coupled with the torso, via first order semi-implicit time-marching schemesinvolving a fully decoupled computation of the unknown fields (ionic state, trans-membrane potential, extracellular and torso potentials). For the isolated bidomainsystem, we show that the Gauss-Seidel and Jacobi like splittings do not compro-mise energy stability; they simply alter the energy norm. Time-step constraints areonly due to the semi-implicit treatment of the non-linear reaction terms. Within theframework of the numerical simulation of electrocardiograms (ECG), these bidomainsplittings are combined with an explicit Robin-Robin treatment of the heart-torsocoupling conditions. We show that the resulting schemes allow a fully decoupled(energy) stable computation of the heart and torso fields, under an additional mildCFL like condition. Numerical simulations, based on anatomical heart and torsogeometries, illustrate the stability and accuracy of the proposed schemes
153
Parallel Preconditioners for Solving Fluid-Structure InteractionsProblems in Hemodynamics
Session Schedule Author Index Session Index M12
Gwenol GrandperrinCMCS, EPFL, [email protected]
Paolo CrosettoCMCS, EPFL, [email protected]
Simone DeparisCMCS, EPFL, [email protected]
Alfio QuarteroniCMCS, EPFL, [email protected]
AbstractModeling Fluid-Structure Interaction (FSI) in the vascular system is mandatory toreliably compute flow indicators when the vessels undergo large deformations. Theresolution of the fully 3D FSI problem is very expensive; in order to lower the timeto solution and to address complex problems, a parallel framework is necessary. Toachieve good performances on large scale parallel architectures, we have developedpreconditioners for the fully coupled FSI system.The important factors to measure parallel performances of a preconditioner arethe independence on the number of iterations on the cpu count (scalability ofthe preconditioner), on the mesh size (optimality), and on the physical parame-ters (robustness), as well as the strong and weak scalability. We aim at devisingspecific preconditioners for High Performance Computing (HPC). In particular wetake advantage of state of the art preconditioners for Navier-Stokes problems suchas the Pressure-Convection-Diffusion (PCD) preconditioner introduced by Elman,Sylvester, and Wathen to solve efficiently the fluid part of the FSI model.We compare the evolution of the number of iterations to solve the full system withclassical methods on a physiological geometry. We also investigate the strong scal-ability of our FSI solver. All the computations are carried out using the open sourcefinite element library LifeV (www.lifev.org) based on Trilinos (http://trilinos.sandia.gov).
154
Mini Symposium M13Robust Multilevel Methods for Multiscale Prob-lemsSchedule Author Index Session Index
Organizers: Thomas Dufaud, Johannes Kraus, Clemens Pechstein,Robert Scheichl, Jorg Willems
AbstractTypical multiscale problems are PDEs with large variation in the coefficients (inparticular high contrast). The probably “simplest” example is the scalar ellip-tic equation −div(α∇u) = f with a uniformly elliptic coefficient α that can varyover several orders of magnitude throughout the domain. Other important exam-ples are flows in heterogeneous porous media (used in oil reservoir or groundwaterflow simulation) or problems in structural mechanics with heterogeneous, possiblyanisotropic materials. Iterative solvers for large-scale problems of this type need notonly be robust with respect to the discretization parameters, but also with respectto the coefficient heterogeneities, which is in general a difficult task. While thereare several approaches yielding robust–or almost robust–performance in practicalcomputations, the rigorous verification of this robustness is still an open problem inmany situations. The goal of this minisymposium is to collect the state of the art intwo- and multilevel solvers for multiscale problems and to explore the links to nu-merical upscaling. Special emphasis shall be put on how preconditioners, smoothers,coarse spaces, and upscaled equations are constructed (and adapted to the problemof interest) in order to gain robustness – both theoretically and numerically.
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M13P1 Schedule
Robust Multilevel Methods for Multiscale Problems
Schedule Author Index Session Index
Date: Monday, June 25Time: 16:00-18:10Location: PetriChairman: Thomas Dufaud, Johannes Kraus, Clemens Pechstein,Robert Scheichl, Jorg Willems
16:05-16:30 : Victorita DoleanAnalysis of Two-Level Method for Heterogeneous Darcy Equation based onLocal Dirichlet to Neumann MapsAbstract
16:30-16:55 : Nicole SpillaneGenEO: A Coarse Space based on Generalized Eigenvalue Problems in theOverlapsAbstract
16:55-17:20 : Jinchao XuSingle-Grid Multilevel MethodAbstract
17:20-17:45 : Juan GalvisMultiscale Spectral AMGe Solvers for High-Contrast Flow ProblemsAbstract
17:45-18:10 : Clark DohrmannConstraint and Weight Selection Algorithms for BDDCAbstract
M13 Abstract
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156
Analysis of Two-Level Method for Heterogeneous DarcyEquation based on Local Dirichlet to Neumann Maps
Session Schedule Author Index Session Index M13
Victorita DoleanUniversite de Nice and Universite de [email protected]
Frederic NatafUniversite Pierre et Marie Curie (Paris VI)[email protected]
Robert ScheichlUniversity of [email protected]
Nicole SpillaneUniversite Pierre et Marie Curie (Paris VI)[email protected]
AbstractCoarse grid correction is a key ingredient in order to have scalable domain decom-position methods. For smooth problems, the theory and practice of such two-levelmethods is well established, but this is not the case for problems with complicatedvariation and high contrasts in the coefficients. We present here a rigorous analy-sis of a two-level overlapping additive Schwarz method (ASM) with a coarse spacebased on low frequency modes of local subdomains Dirichlet-to-Neumann (DtN)maps. We also provide an automatic criterion for the number of modes that needto be added per subdomain to obtain a convergence rate of the order of the con-stant coefficient case. Our method is suitable for parallel implementation and itsefficiency is demonstrated by numerical examples on some challenging problemswith high heterogeneities for automatic partitionings.
157
GenEO: A Coarse Space based on Generalized EigenvalueProblems in the Overlaps
Session Schedule Author Index Session Index M13
Nicole SpillaneUniversite Pierre et Marie Curie (Paris VI)[email protected]
Victorita DoleanUniversity of Nice and University of [email protected]
Patrice HauretManufacture francaise des pneus [email protected]
Frederic NatafUniversite Pierre et Marie Curie (Paris VI)[email protected]
Clemens PechsteinJohannes Kepler [email protected]
Robert ScheichlUniversity of [email protected]
AbstractWe take advantage of the pre-existing framework for the study of two-level addi-tive Schwarz preconditioners to build a coarse space which is robust with regardto heterogeneities in any of the coefficients in the PDEs. We do this regardlessof the way the domain is split into subdomains.In order to identify which compo-nents of the solution should be part of the coarse space we solve local generalizedeigenproblems in each subdomain. In fact, the eigenproblems are defined only onthe part of each subdomain which is overlapped by neighbouring subdomains. Thisallows us to identify a family of low-frequency modes which slow down convergence.Our method is suitable for parallel implementation and its implementation onlyrequires the knowledge of the element matrices. We give a rigorous theoretical re-sult for the condition number of the two-level overlapping additive Schwarz methodwith this coarse space and demonstrate its efficiency through numerical exampleson some challenging three dimensional problems with high heterogeneities for au-tomatic partitions.
158
Single-Grid Multilevel Method
Session Schedule Author Index Session Index M13
Jinchao XuPennsylvania State University, University Park, PA, [email protected]
AbstractIn this talk, I will present a new approach to designing algebraic multigrid methodsfor discretized PDEs discretized on general unstructured grids. The main issue toaddress is parallelization.
159
Multiscale Spectral AMGe Solvers for High-Contrast FlowProblems
Session Schedule Author Index Session Index M13
Juan GalvisDepartment of Mathematics and Institute for Scientific Computations, Texas A&MUniversity, College Station, TX [email protected]
Yalchin EfendievDepartment of Mathematics and Institute for Scientific Computations, Texas A&MUniversity, College Station, TX [email protected]
Panayot S. VassilevskiCenter for Applied Scientific Computing, Lawrence Livermore National Laboratory,P.O. Box 808, L-561, Livermore, CA 94551, [email protected]
AbstractWe construct and analyze multigrid methods with nested coarse spaces for second-order elliptic problems with high-contrast multiscale coefficients. The design of themethods utilizes stable multilevel decompositions with a bound that generally growswith the number of levels. To stabilize this growth, in our theory, we use AMLI-cycle multigrid which leads to an overall optimal cost algorithm. The robustness,with respect to the contrast, is guaranteed due to the combined effect of the Schwarzsmoothers used and the spectral construction of the coarse bases. More specifically,in order to obtain an optimal multilevel decomposition, we combine multigrid ideasin the recent two-level methods in [Multiscale Model. Simul. 8(5), 1621-1644],and earlier, in the element-based algebraic multigrid methods (or AMGe), that uselocal spectral problems to enrich the coarse space. In general, the intermediatecoarse spaces need to be enriched in order to get contrast-independent convergence.The general techniques presented here allow us to study the problem of an optimalenrichment in the sense of enriching with a minimal number of extra coarse degreesof freedom. Thus, the methods we develop are optimal, with respect to both thecontrast and the number of levels used. Moreover, we have the potential to achievethis goal with a minimal number of coarse degrees of freedom. We present numericalresults that illustrate our theoretical findings. This work was performed underthe auspices of the U.S. Department of Energy by Lawrence Livermore NationalLaboratory under Contract DE-AC52-07NA27344.
160
Constraint and Weight Selection Algorithms for BDDC
Session Schedule Author Index Session Index M13
Clark DohrmannSandia National Laboratories, Albuquerque, New Mexico, [email protected]
Clemens PechsteinJohannes Kepler University, Linz, [email protected]
AbstractWe present constraint and weight selections algorithms for Balancing Domain De-composition by Constraints (BDDC) which can be used to design preconditionersof known quality. Both algorithms are motivated by the goal to minimize a certaincondition number estimate that only requires local subdomain information. Numer-ical examples are presented for a variety of different problems to confirm the theoryand to demonstrate the utility of the algorithms. These problems include subdo-mains with two or more materials, almost incompressible elasticity, and subdomainswith irregular boundaries.
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M13P2 Schedule
Robust Multilevel Methods for Multiscale Problems
Schedule Author Index Session Index
Date: Tuesday, June 26Time: 10:30-12:15Location: PetriChairman: Thomas Dufaud, Johannes Kraus, Clemens Pechstein,Robert Scheichl, Jorg Willems
10:35-11:00 : Petr VanekAn Alternative to Domain Decomposition Methods based on PolynomialSmoothingAbstract
11:00-11:25 : Robert ScheichlEnergy Minimizing Coarse Space ConstructionAbstract
11:25-11:50 : James BrannickRecent Advances in Algebraic MultigridAbstract
11:50-12:15 : Marco BuckDomain Decomposition Preconditioners for the Multiscale Analysis of LinearElastic CompositesAbstract
M13 Abstract
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162
An Alternative to Domain Decomposition Methods based onPolynomial Smoothing
Session Schedule Author Index Session Index M13
Petr VanekDepartment of Mathematics, University of West Bohemia, Plzen, Czech Republicaddress: Univerzitnı 22, 306 [email protected]
AbstractThe domain decomposition methods are efficient tools for solving large–scale linearsystems that originate from discretizing elliptic partial differential equations. Ingeneral, the computational domain is decomposed into subdomains and the globalproblem is solved by means of subdomain solvers. The typical domain decomposi-tion method is a variational two–level method with a coarse–space of a resolutionfollowing closely the subdomain size and a massive smoother based on local solvers.This global frame is suited well for both distributed memory and shared memoryarchitectures. The majority of a computational work is done by subdomain solvers;each subdomain solver is assigned to a single processor. This computational organi-zation results in parallelism that uses the number of processors equal to the numberof subdomains. Recently, it became popular to use massively parallel graphic cardsin high–performance computing. Contemporary graphic cards are basically vectorprocessors. Hence, it is desirable to design domain decomposition type methods thatare based on the action y = Ax as a key operation. Indeed, this action allows touse up to n = ord(A) processors and can be performed very efficiently on a vector–type architecture. We propose a class of methods based on polynomial smoothingrather than subdomain solvers. The key operation is the action of a massive poly-nomial smoother with an error propagation operator E = (I−α1A). . . . .(I−αNA).Those methods are generally variational multigrid methods with aggressive coars-ening and massive polynomial smoothing. For those methods, we prove an optimalconvergence result (independent of both fine and coarse–level resolution) providedthe degree of our polynomial smoother (number of smoothing Richardson sweeps)is about 1/2H/h (h is the fine and H is the coarse–level resolution). The resultingalgorithms are (even in serial case) asymptotically much cheaper than domain de-composition methods based on subdomain solvers and allow for massive parallelismon vector machines.
163
Energy Minimizing Coarse Space Construction
Session Schedule Author Index Session Index M13
Robert ScheichlUniversity of Bath, United [email protected]
Ivan G. GrahamUniversity of Bath, United [email protected]
Ludmil ZikatanovPenn State University, United [email protected]
AbstractEnergy minimizing coarse spaces are key to robust multilevel iterative methods formultiscale elliptic problems. They are at the heart of algebraic multigrid methodsand have been extensively analysed over the last 5 years and extended also to sys-tems of partial differential equations. How exactly to minimise the energy in aneffective way, while still maintaining sparsity and scalability with a rigorous under-lying analysis, is still an ongoing international quest. In this talk we would like toreturn to a simple algebraic method that we presented a few years ago, in the con-text of two-level additive Schwarz. Given a set of supports, this method finds sucha minimum energy basis subject to a partition of unity constraint using one localsolve per coarse space basis function and one global solve to enforce the partitionof unity constraint. Although this global solve may seem prohibitively expensive,we argued then that a one-level overlapping Schwarz method is an effective andscalable preconditioner and showed that such a preconditioner can be implementedefficiently using the Sherman-Morrison-Woodbury formula. The result was an ele-gant, scalable, algebraic method for constructing a robust coarse space given onlythe supports of the coarse space basis functions. Numerical experiments confirmedthis. We now present an analysis of this coarse space construction that rigorouslyjustifies its optimal complexity. Numerical experiments also show that, when usedin a two-level preconditioner, the energy minimizing coarse space gives better re-sults than some other coarse space constructions. We finish with some commentson how to extend the ideas to energy minimizing coarse spaces with more than oneconstraint.
164
Recent Advances in Algebraic Multigrid
Session Schedule Author Index Session Index M13
James BrannickPenn State [email protected]
AbstractIn this talk, I will highlight several recent advances in the development and analysisof AMG coarsening algorithms. I will discuss various strategies for selecting thecoarse variables and defining interpolation, in both the classical AMG and matching(aggregation) AMG settings. Numerical experiments of the proposed techniquesapplied to various challenging linear systems will also be provided.
165
Domain Decomposition Preconditioners for the MultiscaleAnalysis of Linear Elastic Composites
Session Schedule Author Index Session Index M13
Marco BuckFraunhofer Institute for Industrial Mathematics (ITWM), Kaiserslautern, Germany;University of [email protected]
Oleg IlievFraunhofer Institute for Industrial Mathematics (ITWM), Kaiserslautern, [email protected]
Heiko AndraFraunhofer Institute for Industrial Mathematics (ITWM), Kaiserslautern, [email protected]
AbstractWe analyse two-level overlapping Schwarz domain decomposition methods for afinite element discretization of the PDE system of linear elasticity. The focus inour study lies in the application to compressible, particle-reinforced composites in3D with large jumps in their material coefficients. We present coefficient-explicitbounds for the condition number of the two-level Additive Schwarz preconditionedlinear system. Thereby, we do not require that the coefficients are resolved by thecoarse mesh. The bounds show a dependence of the condition number on the energyof the coarse basis functions, the coarse mesh and the overlap parameters. Similarestimates have been developed for scalar elliptic PDEs in the work of Graham,Lechner and Scheichl (“Domain decomposition for multiscale PDEs”). The coarsespaces to which they apply here are assumed to contain the six rigid body modes andcan be considered as generalizations of the space of piecewise linear vector valuedfunctions on a coarse triangulation. The developed estimates provide a conceptfor the construction of coarse spaces which can lead to preconditioners which arerobust w.r.t. discontinuities in the Young’s modulus and the Poisson ratio of theunderlying composite.To confirm the theoretical results numerically, we first extend the linear multiscalefinite element method as formulated by T. Hou and X. Wu to the system of linearelasticity. E.g., using a multiscale coarse space and assuming that inclusions ofhigh contrast are isolated in the interior of coarse elements, we observe conditionnumber bounds independent of variations in the Young’s modulus and the Poissonratio. Further on, linear and energy minimizing coarse spaces are discussed.
166
M13P3 Schedule
Robust Multilevel Methods for Multiscale Problems
Schedule Author Index Session Index
Date: Tuesday, June 26Time: 16:00-17:45Location: PetriChairman: Thomas Dufaud, Johannes Kraus, Clemens Pechstein,Robert Scheichl, Jorg Willems
16:05-16:30 : Florian ThominesA Systematic Coarse-Scale Model Reduction Technique forParameter-Dependent Flows in Highly Heterogeneous MediaAbstract
16:30-16:55 : Ivan GrahamMultiscale Finite Elements for High-Contrast Elliptic ProblemsAbstract
16:55-17:20 : Jan NordbottenApproximate Multilevel Solvers for Flow and Transport in Porous MediaAbstract
17:20-17:45 : Xiaozhe HuParallel AMG Method on GPUAbstract
M13 Abstract
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167
A Systematic Coarse-Scale Model Reduction Technique forParameter-Dependent Flows in Highly Heterogeneous Media
Session Schedule Author Index Session Index M13
Florian ThominesEcole des Ponts - ParisTech, Champs-sur-Marne, F-77455 Marne-la-Vallee cedex 2and INRIA, MICMAC project, 78153 Le Chesnay Cedex, [email protected]
Yalchin EfendievDepartment of Mathematics and ISC, Texas A & M University, College Station,TX [email protected]
Juan GalvisDepartment of Mathematics and ISC, Texas A & M University, College Station,TX [email protected]
AbstractThe talk will discuss a multiscale approach for solving the parameter-dependent el-liptic equation with highly heterogeneous coefficients. In particular, we assume thatthe coefficients have both small scales and high contrast (where the high contrastrefers to the large variations in the coefficients). The main idea of our approachis to construct local basis functions that encode the local features present in thecoefficient to approximate the solution of parameter-dependent flow equation. Con-structing local basis functions involves (1) finding initial multiscale basis functionsand (2) constructing local spectral problems for complementing the initial coarsespace. We use the Reduced Basis (RB) approach to construct a reduced dimensionallocal approximation that allows quickly computing the local spectral problem. Thisis done following the RB concept by constructing a low dimensional approximationoffline. For any online parameter value, we use a reduced dimensional approxi-mation of the local problem to construct multiscale basis functions. These localcomputations are fast and are used to solve the coarse-scale dimensional problem.The coarse problem is used to construct robust iterative methods of the domaindecomposition type. Our numerical results show that one can achieve a substantialdimension reduction when solving the local spectral problems. We discuss conver-gence of the method and the computational cost of the proposed method.
168
Multiscale Finite Elements for High-Contrast Elliptic Problems
Session Schedule Author Index Session Index M13
Ivan G. GrahamUniversity of [email protected]
AbstractWe first discuss multiscale finite element methods for elliptic interface problems withhigh contrast coefficients. These are approximated on coarse quasiuniform meshes,which do not need to resolve the interfaces. The methods are H1- conforming, andrequire the solution of subgrid problems for the basis functions on elements whichstraddle the coefficient interface, but use standard linear approximation otherwise.The methods have (optimal) convergence rate in the energy and L2 norms, indepen-dent of the “contrast” (i.e. ratio of largest to smallest value) of the PDE coefficient.A key point is the introduction of coefficient-dependent interior boundary conditionsfor the subgrid problems. Since these boundary conditions are rather technical anddelicate, we also investigate more generally applicable adaptive methods which aimto find appropriate boundary conditions automatically.
169
Approximate Multilevel Solvers for Flow and Transport in PorousMedia
Session Schedule Author Index Session Index M13
Jan Martin NordbottenUniversity of [email protected]
Eirik KeilegavlenUniversity of [email protected]
Tor Harald SandveUniversity of [email protected]
AbstractThe governing equations for flow and transport in porous media are characterizedby non-linear constitutive relationships that are heterogeneous at almost every spa-tial scale. Furthermore, for geological applications, the heterogeneous coefficientsof these constitutive relationships may be highly uncertain.
Thus for practical problems, the parameters of the governing equations are atbest known statistically, and the simulation results must correspondingly be inter-perted to have a significant degree of uncertainty. Compounding this uncertaintyis significant spatial extent of the domains of interest, forcing engineers to applydiscrete spatial and temporal resolutions that cannot be considered converged.
Despite the large uncertainty in the model problem and the approximationsintroduced through discretization, it is common practice to solve the resulting lin-ear and non-linear equations to a high degree of accuracy. This is necessary, asmost numerical algorithms for solving the discrete system of non-linear equationsare sensitive to approximation errors in the solvers.
Our goal is to provide inexact solvers that are in a sense structure preserving,while retaining efficiency. The exact structures to be preserved are dependent onboth the underlying problem and overall numerical algorithm. Examples includemonotonicity or divergence of the approximate solution. The structure preserva-tion thus allows us to apply inexact linear and non-linear solvers without losing therobustness of the numerical algorithm.
In this talk, we present the framework a framework for structure-preservinginexact solvers in the setting on preconditioners for linear systems. From a theo-retical viewpoint, we discuss how structure preservation allows for a priori and aposteriori estimates. From a practical viewpoint, we highlight the realization of theframework for time-dependent flows in fractured porous media. For this applica-tion, the framework allows us to extract coarse-level discretizations, which we useto construct an adaptive coarsening algorithm.
170
Parallel AMG Method on GPU
Session Schedule Author Index Session Index M13
Xiaozhe HuThe Pennsylvania State Universityhu [email protected]
AbstractDeveloping parallel algorithms for solving large sparse linear systems is an importantand challenging task in scientific computing and practical applications. In thiswork, we develop a new parallel algebraic multigrid (AMG) method for GPU. Thecoarsening and smoothing procedures in our new algorithm are based on a regionquadtree (octree in 3D) generated from an auxiliary grid. This provides (nearly)optimal load balance and predictable communication patterns — factors that makeour new algorithm suitable for parallel computing, especially on GPU. Numericalresults show that our new method can speed up the existing GPU code (CUSPfrom NVIDIA) by a factor of 4 on a quasi-uniform grid and by a factor of 2 ona shape-regular grid for certain model problems. This work is co-authored by J.Cohen, L. Wang, and J. Xu.
171
M13P4 Schedule
Robust Multilevel Methods for Multiscale Problems
Schedule Author Index Session Index
Date: Wednesday, June 27Time: 10:30-12:15Location: PetriChairman: Thomas Dufaud, Johannes Kraus, Clemens Pechstein,Robert Scheichl, Jorg Willems
10:35-11:00 : Baptiste PoirriezDeflation and Neumann-Neumann Preconditioner for Schur DomainDecomposition MethodAbstract
11:00-11:25 : Thomas DufaudAn Algebraic Multilevel Preconditioning Framework based on Information ofa Richardson ProcessAbstract
11:25-11:50 : Svetozar MargenovMultilevel Preconditioning of Strongly Anisotropic Elliptic ProblemsAbstract
11:50-12:15 : Johannes KrausRobust Domain Decomposition Multigrid Methods using Additive SchurComplement ApproximationAbstract
M13 Abstract
Part 1
Part 2
Part 3
172
Deflation and Neumann-Neumann Preconditionner for SchurDomain Decomposition Method
Session Schedule Author Index Session Index M13
Baptiste PoirriezIrisa, [email protected]
Jocelyne ErhelInria, [email protected]
AbstractNumerical simulations in fractured media are an essential tool for studying hy-draulic properties. Discrete Fracture Networks are composed of many multiscaleplane fractures intersecting each other, leading to complex geometries. We assumethat the rock surrounding the fractures is impervious and we aim at simulating theflow in the fractures. Governing equations are Darcy’s law and mass continuity,with continuity conditions at the intersections. Mesh generation is rather difficultin this context, because of the geometry, and requires a specific method. We applya Mixed Hybrid Finite element method and get a large sparse symmetric positivedefinite (spd) linear system to solve.We study a domain decomposition method, which takes advantages from boththe direct method and the Preconditioned Conjugate Gradient (PCG). This Schurmethod reduces the global problem to an interface problem, with a natural domaindecomposition based on fractures or fracture packs. We propose an original ap-proach for optimizing the algorithm and a global preconditioning of deflation type.Since the Schur complement S is spd, we apply PCG to solve the linear systemSx = b. We use the classical Neumann-Neumann (NN) preconditioner. To gain inefficiency, we use only one Cholesky factorization of the subdomain matrices for thepreconditionning and the conjugate gradient steps. We also define a coarse space,based on the subdomain definition, to apply a deflation preconditioner.We do a theoretical complexity study of our algorithm. We use this study, withthe numerical data, to compute experimental complexity. We compare the resultsbetween several combination for the preconditioner. Then, we confront our resultswith existing solvers.
173
An Algebraic Multilevel Preconditioning Framework based onInformation of a Richardson Process
Session Schedule Author Index Session Index M13
Thomas DufaudINRIA - Rennes - Bretagne [email protected]
AbstractA fully algebraic framework for constructing coarse spaces for multilevel precondi-tioning techniques is proposed. Multilevel techniques are known to be robust forscalar elliptic Partial Differential Equations with standard discretization and to en-hance the scalability of domain decomposition method such as RAS preconditioningtechniques. An issue is their application to linear system encountered in industrialapplications which can be derived from non-elliptic PDEs. Moreover, the buildingof coarse levels algebraically becomes an issue since the only known information iscontained in the operator to inverse. Considering that a coarse space can be seenas a space to represent an approximated solution of a smaller dimension than theleading dimension of the system, it is possible to build a coarse level based on acoarse representation of the solution. Drawing our inspiration from the Aitken-SVD methodology, dedicated to Schwarz methods, we proposed to construct anapproximation space by computing the Singular Value Decomposition of a set ofiterated solutions of the Richardson process associated to a given preconditioner.This technique does not involve the knowledge of the underlying equations andcan be applied to build coarse levels for several preconditioners. Numerical resultsare provided on both academic and industrial problems, using two-level additivepreconditioners built with this methodology.
174
Multilevel Preconditioning of Strongly Anisotropic EllipticProblems
Session Schedule Author Index Session Index M13
Svetozar MargenovIICT-BAS, Sofia, [email protected]
Johannes KrausRICAM, Linz, [email protected]
Maria LymberyIICT-BAS, Sofia, Bulgariamariq [email protected]
AbstractThe talk commences by giving an overview of the second order elliptic problem dis-cretized by linear conforming or nonconforming Crouzeix-Raviart finite elements(FE). In order to obtain a prescribed accuracy a uniform recursive refinementof the initially introduced triangulation T0 is performed, and the nested meshesT0 ⊂ T1 ⊂ ... ⊂ T` = Th are constructed. The main focus of the study is on thedevelopment of robust multilevel preconditioning methods for strongly anisotropicproblems. The first part is devoted to the construction and analysis of AlgebraicMultiLevel Iteration (AMLI) methods in the case of coefficient jumps which arealigned with the interfaces of the initial mesh T0. The presented condition numberestimates are uniform with respect to both mesh and/or coefficient anisotropy, thejumps, as well as the size of the discrete problem. The case of higher order FEsis discussed in the second part of the talk. For instance, for quadratic FEs thestandard hierarchical basis techniques do not result in splittings in which the an-gle between the coarse space and its hierarchical complement is uniformly boundedwith respect to the anisotropy ratio. Here some recent alternative results are pre-sented based on additive Schur complement approximations, including the case ofanisotropy which is not aligned with the mesh. The two-level method is recursivelyapplied in the construction of robust multilevel preconditioners with optimal ornearly optimal order of computational complexity.
175
Robust Domain Decomposition Multigrid Methods usingAdditive Schur Complement Approximation
Session Schedule Author Index Session Index M13
Johannes KrausRICAM, Linz, [email protected]
AbstractSparse Schur complement approximations play a key role in various iterative meth-ods for solving systems of linear algebraic equations arising from finite elementdiscretization of partial differential equations. In this talk we consider an algorithmfor additive Schur complement approximation that is based on computing and as-sembling exact Schur complements of local (stiffness) matrices associated with acovering of the entire domain by overlapping subdomains. The resulting coarse-grid matrix is sparse and is shown to be spectrally equivalent to the (global) Schurcomplement with a bound on the relative condition number independent of thevariations in the coefficients of the model elliptic equation. This approach allowsfor constructing a variational multigrid method that provides energy minimizinginterpolation on an auxiliary space. The related two-grid method is analyzed us-ing the fictitious space lemma. Several possibilities of exploiting this new type ofcoarse-grid operator are illustrated. Numerical experiments demonstrate uniformmultigrid convergence for problems with highly oscillatory coefficients.
176
Mini Symposium M14100% Parallelizable Algorithms for Symmet-ric, Indefinite and Non-Symmetric Problems
Schedule Author Index Session Index
Organizers: Ismael Herrera and Luis Miguel de la Cruz
AbstractDomain decomposition methods (DDM) are the most efficient means for applyingparallel-computing to the solution of partial differential equations. Thus, a maingoal of DDM research has been to develop highly parallelizable algorithms. Asa result, since the international community began intensively studying DDM, at-tention has shifted from overlapping to non-overlapping methods, mainly becausealgorithms derived from non-overlapping methods can achieve a higher level of par-allelization. Furthermore, the impressive progress of parallel hardware that hastaken place in recent years demands the availability of 100%-in-parallel-software.On the other hand, at present it is recognized that competitive algorithms need toincorporate constraints, such as continuity on primal-nodes. This, however, posesa new challenge for developing 100% parallelizable algorithms, which has been dif-ficult to overcome. This mini-symposium is devoted to present and discuss fouralgorithms with constraints of wide applicability, all of them 100% parallelizable.Such algorithms can be applied to symmetric, non-symmetric and indefinite prob-lems. The new algorithms have been derived in the realm of a framework recentlyintroduced: the DVS-framework [1-3]. Two of them are the DVS versions of BDDCand FETI-DP, respectively, while we could not identify in the literature algorithmswith clear similarities to the other two. This mini-symposium is made of four lec-tures, two of them devoted to explain the new algorithms in general and the othertwo to applications.
REFERENCES
[1] Herrera, I. et al. Geofisica Internacional, 50, pp 445-463, 2011.[2] Herrera, I. et al. NUMER. METH. PART D. E. 27, pp. 1262-1289, 2011.[3] Herrera, I. et al. NUMER. METH. PART D. E. 26, pp. 874-905, 2010.
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M14 Schedule
100% Parallelizable Algorithms for Symmetric, Indefi-nite and Non-Symmetric Problems
Schedule Author Index Session Index
Date: Monday, June 25Time: 10:30-12:15Location: PetriChairman: Ismael Herrera and Luis Miguel de la Cruz
10:35-11:00 : Ismael HerreraFour Massively Parallel Algorithms for Symmetric, Indefinite andNon-Symmetric Matrices: OverviewAbstract
11:00-11:25 : Luis Miguel de la CruzFour Massively Parallel Algorithms for Symmetric, Indefinite andNon-Symmetric Matrices: Implementation IssuesAbstract
11:25-11:50 : Alberto RosasFour Massively Parallel Algorithms for Symmetric, Indefinite andNon-Symmetric Matrices: Applications to a Single EquationAbstract
11:50-12:15 : Ivan ContrerasFour Massively Parallel Algorithms for Static ElasticityAbstract
178
Four Massively Parallel Algorithms for Symmetric, Indefinite andNon-Symmetric Matrices: Overview
Session Schedule Author Index Session Index M14
Ismael HerreraInstituto de Geofısica Universidad Nacional Autonoma de [email protected]
AbstractIn this lecture a set of four general purpose algorithms that are very suitable forbuilding the software that is required for efficiently programming the most pow-erful parallel computers available at present are introduced and explained. Themost effective procedures for approaching the ideal of achieving totally parallelizedalgorithms are derived from non-overlapping domain decomposition methods. Inprinciple, the goal is to develop algorithms capable of constructing the global solu-tion by solving local problems, in each partition subdomain, exclusively. At present,however, competitive algorithms need to incorporate constraints, such as continuityon primal-nodes. This poses an additional challenge, which has been difficult toovercome. This mini-symposium is devoted to present and discuss four general pre-conditioned algorithms with constraints in which such a goal is achieved; namely, theglobal solution is obtained by solving local problems exclusively. Each one of suchfour algorithms can be applied to symmetric, non-symmetric and indefinite prob-lems. The new algorithms have been derived in the realm of the DVS-framework,recently introduced [1-3]. Two of them are the DVS versions of BDDC and FETI-DP, respectively. As for the other two, we have not identified in the literature al-gorithms with clear similarities to them. Keywords: Massively-parallel algorithms;parallel-computers; non-overlapping DDM; DDM with constraints; BDDC; FETI-DP
[1] Herrera, I., Carrillo-Ledesma A. & Rosas-Medina A. A Brief Overview of Non-overlapping Domain Decomposition Methods, Geofisica Internacional, Vol. 50(4),pp 445-463, 2011.
[2] Herrera, I. & Yates R. A. The Multipliers-Free Dual Primal Domain Decompo-sition Methods for Nonsymmetric Matrices, NUMER. METH. PART D. E. 27(5)pp. 1262-1289, 2011. DOI 10.1002/Num. 20581. (Published on line April 28, 2010)
[3] Herrera, I. & Yates R. A. The Multipliers-free Domain Decomposition Methods,NUMER. METH. PART D. E. 26(4) pp. 874-905, 2010, DOI 10.1002/num. 20462(Published on line April 23, 2009).
179
Four Massively Parallel Algorithms for Symmetric, Indefinite andNon-Symmetric Matrices: Implementation Issues
Session Schedule Author Index Session Index M14
Luis Miguel de la CruzGeophysics Institute, Universidad Nacional Autonoma de [email protected]
AbstractNowadays parallel computing is ubiquitous and almost all new computational re-sources contain more than one processing unit. This outstanding progress gives usthe opportunity to develop parallel codes that take advantage of the current paral-lel architectures. On the other hand, domain decomposition methods (DDM) allowus to model macroscopic systems applying efective parallel algorithms, and in par-ticular, the non-overlapping methods can achieve a higher level of parallelization.The first talk of this mini-symposium was devoted to present a set of four generalpurpose algorithms that are very suitable for efficiently programming the powerfulparallel computers available at present. These new algorithms have been derived inthe realm of the DVS-framework [1-3]. The numerical and computational issues, aswell as some examples of application are presented in this talk. From the point ofview of software engineering, the DVS-framework offers a general plataform whichgive us a natural separation of the concepts and operations, that results in gen-eral, efficient and elegant codes. In this implementation we use the Finite VolumeMethod (FVM) to obtain the numerical model, although we could have used almostany other discretization procedure. We apply object oriented and generic program-ming paradigms in order to generate several generic units that in turn can be usedto construct the codes for the algorithms of DVS. These algorithms are iterative innature, and are based on some well known Krylov methods, to say CGM, GMRESor some others. Finally, we present some parallelization metrics that measure thespeedup and efficiency of our implementations.
[1] Herrera, I. et al. Geofisica Internacional, 50, pp 445-463, 2011.
[2] Herrera, I. et al. NUMER. METH. PART D. E. 27, pp. 1262-1289, 2011.
[3] Herrera, I. et al. NUMER. METH. PART D. E. 26, pp. 874-905, 2010.
180
Four Massively Parallel Algorithms for Symmetric, Indefinite andNon-Symmetric Matrices: Applications to a Single Equation
Session Schedule Author Index Session Index M14
Alberto RosasPosgrado en Ciencias de la Tierra Universidad Nacional Autonoma de [email protected]
Ismael HerreraInstituto de Geofısica Universidad Nacional Autonoma de [email protected]
AbstractIn the introductory lecture of this minisymposium a set of four general purposealgorithms that are very suitable for building the software that is required for ef-ficiently programming the most powerful parallel computers available at presentwere introduced and explained. Such algorithms are equally applicable to a singlepartial-differential equation and to systems made of many such equations; thus, inthis mini-symposium that is devoted to them, applications are dealt with in somedetail in two lectures: the present one, in which single-equation applications are dis-cussed and another one in which system-of-equations applications are treated. Allthe algorithms considered were derived from non-overlapping domain decomposi-tion methods with constraints using the derived-vector space (DVS) framework,recently introduced [1-3], and are applicable to symmetric, indefinite and non-symmetric matrices. The feature, shared by they all, that permits making themmassively-parallel is that the global solution is obtained by solving local problems,in each partition subdomain, exclusively. Keywords: Massively-parallel algorithms;parallel-computers; non-overlapping DDM; DDM with constraints; BDDC; FETI-DP
[1] Herrera, I., Carrillo-Ledesma A. & Rosas-Medina A. A Brief Overview of Non-overlapping Domain Decomposition Methods, Geofisica Internacional, Vol. 50(4),pp 445-463, 2011.
[2] Herrera, I. & Yates R. A. The Multipliers-Free Dual Primal Domain Decompo-sition Methods for Nonsymmetric Matrices, NUMER. METH. PART D. E. 27(5)pp. 1262-1289, 2011. DOI 10.1002/Num. 20581. (Published on line April 28, 2010)
[3] Herrera, I. & Yates R. A. The Multipliers-free Domain Decomposition Methods,NUMER. METH. PART D. E. 26(4) pp. 874-905, 2010, DOI 10.1002/num. 20462(Published on line April 23, 2009).
181
Four Massively Parallel Algorithms for Static Elasticity
Session Schedule Author Index Session Index M14
Ivan ContrerasPosgrado en Ciencia de la Computacion Universidad Nacional Autonoma de [email protected]
Ismael HerreraInstituto de Geofısica Universidad Nacional Autonoma de [email protected]
AbstractIn order to profit from the parallel hardware available nowadays, massively-parallelizedhardware is required. There are bounds for the level of parallelization that can beactually achieved beyond which it is not possible to go. For non-overlapping domaindecompositions the goal is to develop algorithms capable of constructing the globalsolution by solving local problems, in each partition subdomain, exclusively. It hasbeen generally recognized that for this purpose the introduction of constraints, asit is required today by competitive algorithms, constitutes an additional difficultynot easy to overcome. Fortunately, a set of four general purpose algorithms withconstraints possessing such a feature and applicable to a broad class of matrices-symmetric, indefinite and non- symmetric- has recently been developed, as it isexplained in the introductory lecture of this minisymposium. Thus, in the presenttalk we announce and explain four massively-parallel algorithms that have been de-rived from them for specifically treating the system of equations that govern StaticElasticity. All this work has been carried out in the derived-vector space framework(DVS-framework), recently introduced by I. Herrera and his co-workers [1-3]. Key-words: Massively-parallel algorithms; parallel-computers; non-overlapping DDM;DDM with constraints; elasticity; BDDC; FETI-DP
[1] Herrera, I. et al. Geofisica Internacional, 50, pp 445-463, 2011.
[2] Herrera, I. et al. NUMER. METH. PART D. E. 27, pp. 1262-1289, 2011.
[3] Herrera, I. et al. NUMER. METH. PART D. E. 26, pp. 874-905, 2010.
182
Mini Symposium M15Space-Time Parallel MethodsSchedule Author Index Session Index
Organizers: Martin J. Gander, Felix Kwok and Yvon Maday
AbstractOften problems from real world applications are time dependent, and classical timestepping methods lend themselves only to parallelization in space. With the ad-vent of new generation parallel computers with hundred-thousands of cores, thespace direction is easily saturated, and one needs to include the time direction forparallelization. This minisymposium consists of recent research results on space-time parallel methods, with major new contributions like Dirichlet-Neumann andNeumann-Neumann methods for evolution problems, the application of the pararealalgorithm to hyperbolic systems of conservation laws, Hamiltonian problems, spec-tral deferred correction variants of the parareal algorithm, and also new results forSchwarz waveform relaxation methods, moving mesh methods and control.
Part 1
Part 2
Part 3
183
M15P1 Schedule
Space-Time Parallel Methods
Schedule Author Index Session Index
Date: Tuesday, June 26Time: 10:30-12:15Location: MarkovChairman: Martin J. Gander, Felix Kwok and Yvon Maday
10:35-11:00 : Yvon MadayParareal in Time Algorithm for Hyperbolic SystemsAbstract
11:00-11:25 : Michael MinionEfficient Implementation of a Multi-Level Parallel in Time AlgorithmAbstract
11:25-11:50 : Rim GuetatCoupling Parareal Algorithm with Domain Decomposition MethodsAbstract
11:50:12:15 : Felix KwokNeumann-Neumann Waveform Relaxation Methods for the Time-DependentHeat EquationAbstract
M15 Abstract
Part 2
Part 3
184
Parareal in Time Algorithm for Hyperbolic Systems
Session Schedule Author Index Session Index M15
Yvon MadayUPMC Univ Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions, Paris, FranceDivision of Applied Mathematics, Brown University, Providence, RI, [email protected]
Xiaoying DaiLSEC, Chinese Academy of Sciences, Beijing 100190, [email protected]
AbstractThe parareal in time algorithm allows to perform parallel simulations of time de-pendent problems. This algorithm has been implemented on many types of timedependent problems with some success. Recent contributions have allowed to ex-tend the domain of application of the parareal in time algorithm so as to handle longtime simulations of Hamiltonian systems. This improvement has managed to avoidthe fatal large lack of accuracy of the plain parareal in time algorithm consequenceof the fact that the plain parareal in time algorithm does not conserve invariants.A somehow similar difficulty occurs for problems where the solution lacks regular-ity, either initially or in the evolution, like for the solution to hyperbolic system ofconservation laws. In this paper we identify the problem of lack of stability of theparareal in time algorithm and propose a simple way to cure it. The new method isused to solve a linear wave equation and a non linear Burger’s equation, the resultsillustrate the stability of this variant of the parareal in time algorithm.
185
Efficient Implementation of a Multi-Level Parallel in TimeAlgorithm
Session Schedule Author Index Session Index M15
Michael MinionUniversity of North [email protected]
Matthew EmmettUniversity of North [email protected]
AbstractWe explain how the communication between processors in a multi-level parallel-in-time algorithm for PDEs can be scheduled to reduce blocking communication. Theparticular time-parallel method examined is the parallel full approximation schemein space and time (PFASST), which utilizes a heirarchy of spatial and temporal dis-cretization levels. By decomposing the update to initial conditions passed betweenprocessors into multiple spatial resolutions, the communication at the finest levelcan be scheduled to overlap with computation at coarser levels. We demonstratethe cost savings with a three dimensional PDE example.
186
Coupling Parareal Algorithm with Domain DecompositionMethods
Session Schedule Author Index Session Index M15
Rim GuetatLaboratoire Jacques-Louis Lions, Universite Pierre & Marie Curie, [email protected]
Yvon MadayUPMC Univ Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions, Paris, FranceDivision of Applied Mathematics, Brown University, Providence, RI, [email protected]
Frederic MagoulesApplied Mathematics and Systems Laboratory, Ecole Centrale Paris, [email protected]
AbstractNumerical temporal evolution schemes are sequential in nature and thus have beenviewed as providing limited parallel performance gains. Parallelization with respectto the time variable appears as a seductive alternative. For solving temporal evolu-tion problems using domain decomposition methods, the classical approach consistsof using a parallel solver to solve the equations in space of an implicit time schemeat each time step. To avoid the major drawback of this approach, i.e., solving astationary problem at each time step, an original method, called the parareal algo-rithm has been introduced a decade ago. This algorithm splits the global temporalevolution problem into a series of independent evolution problems on smaller timeintervals. Both a coarse and fine grids have been introduced for the solution intime and the parareal algorithm can easily be summarized as a predictor-correctormethod that allows to get parallelization through the time. For complex prob-lems involving large size geometries, the parareal algorithm suffers from the size ofthe spatial problems to solve. In this paper, we propose to combine the pararealalgorithm and the domain decomposition methods. We demonstrate that the com-bination of these two approaches (decomposition in space and decomposition intime) leads to an effective and robust algorithm, reducing significantly the com-putational time. The coupling parareal algorithm and the domain decompositionmethod have been implemented in our proper library within the C++ language. Nu-merical experiments illustrate the performance of this library on massive parallelcomputers. Applications on realistic industrial problems illustrate the extraordi-nary convergence properties (computational time, speed-up, scalability) of this newmethod.
187
Neumann-Neumann Waveform Relaxation Methods for theTime-Dependent Heat Equation
Session Schedule Author Index Session Index M15
Felix KwokUniversity of [email protected]
Martin J. GanderUniversity of [email protected]
Bankim Chandra MandalUniversity of [email protected]
AbstractWe propose a waveform relaxation version of the Neumann–Neumann method forparabolic problems. Just like for the steady case, one step of the method consists ofsolving the subdomain problems using Dirichlet traces, followed by a correction stepinvolving Neumann interface conditions. However, each subdomain problem is nowin both space and time, and the interface data to be exchanged are also functionsof time. Using a Laplace transform argument, we show for the heat equation thatwhen we consider finite time intervals, the Neumann-Neumann method convergessuperlinearly both in one spatial dimension and for 2D decompositions into strips.The convergence rate depends on T/H2, where T is the length of the time windowand H is the size of the subdomain.
188
M15P2 Schedule
Space-Time Parallel Methods
Schedule Author Index Session Index
Date: Tuesday, June 26Time: 16:00-17:45Location: MarkovChairman: Martin J. Gander, Felix Kwok and Yvon Maday
16:05-16:30 : Stefan GuttelOn the Convergence of Parallel Deferred Correction MethodsAbstract
16:30-16:55 : Martin J. GanderAnalysis of the Parareal Algorithm and a Symmetrized Variant forHamiltonian ProblemsAbstract
16:55-17:20 : Jacques LaskarTime-Parallel Integrations for Long Term Solar System StudiesAbstract
17:20-17:45 : Julien SalomonTime-Parallelization and Optimal Control for NMRAbstract
M15 Abstract
Part 1
Part 3
189
On the Convergence of Parallel Deferred Correction Methods
Session Schedule Author Index Session Index M15
Stefan GuttelUniversity of [email protected]
Martin J. GanderUniversity of [email protected]
AbstractWe discuss some preliminary convergence results for parallel deferred correctionmethods, with an emphasis on the Parareal-SDC variant proposed by Minion &Williams (2008) and Minion (2011), and a novel variant based on barycentric ra-tional interpolation.
190
Analysis of the Parareal Algorithm and a Symmetrized Variantfor Hamiltonian Problems
Session Schedule Author Index Session Index M15
Martin J. GanderUniversity of [email protected]
Ernst HairerUniversity of [email protected]
AbstractHamiltonian problems have many geometric properties, like energy preservation andsymplectic flows. When integrating such problems numerically, it is desirable to usenumerical integrators that preserve some of these properties, in order to get goodapproximations over long time intervals. There are currently many well establishedgeometric integrators, but when they are combined with the parareal algorithm,all good geometric properties are lost. We present a backward error analysis forthe parareal algorithm and a recent symmetrized variant, which shows that indeedthe length of the time interval on which the parareal algorithm is efficient has aprecise bound, which unfortunately also holds for the symmetrized variant. We thenoptimize the parameters in the parareal algorithm for performance when applied toHamiltonian problems, and present numerical results illustrating our analysis.
191
Time-Parallel Integrations for Long Term Solar System Studies
Session Schedule Author Index Session Index M15
Jacques LaskarAstronomie et Systemes Dynamiques, IMCCE, Observatoire de [email protected]
Hugo Jimenez-PerezAstronomie et Systemes Dynamiques, IMCCE, Observatoire de [email protected]
AbstractIn 1997, Saha Stadel and Tremaine have introduced a time-parallel algorithm foralmost integrable Hamiltonian systems applied to the Solar System dynamics. Suchan algorithm is based in an iterative predictor-corrector scheme and the perturba-tion theory for Hamiltonian systems. We remark that they have used a secondorder symplectic method as underlying integrator. In a slightly different approach,in 2001 Lions, Maday and Turinici have introduced another time-parallel schemefor PDEs knew as the ”Parareal” scheme, which is independent of the numericalscheme. With the advent of the GPU computing systems, we have recently de-veloped an extension of the Saha-Stadel-Tremaine algorithm which uses high ordersymplectic integrators taking advantage of the performance of the shared memoryGPU technology. In this talk we explain the details of the algorithm and discusssome results about the speed-up we can reach. We propose an hybrid algorithmwhich combines the pure parareal scheme and our extended algorithm to simulatelong term Solar System dynamics and we will discuss preliminary results for inner,and outer Solar System simulations.
192
Time-Parallelization and Optimal Control for NMR
Session Schedule Author Index Session Index M15
Julien SalomonCEREMADE, Univ. Paris-Dauphine, pl. du Mal. de Lattre de Tassigny, 75016Paris, [email protected]
Mohamed Kamel RiahiCEREMADE, Univ. Paris-Dauphine, pl. du Mal. de Lattre de Tassigny, 75016Paris, [email protected]
Dominique SugnyICB, University of Bourgogne, Dijon, [email protected]
AbstractFrom the mathematical point of view, Nuclear Magnetic Resonance (NMR) hasthe advantage that the results obtained in the simulations can be directly testedexperimentally. Numerical methods play consequently a significant role in the de-sign of efficient magnetic fields. After a short description of the model used inNMR, the Block equations, we will present an algorithm based on a relevant timeparallelization that enables us to compute rapidly optimal magnetic fields.
193
M15P3 Schedule
Space-Time Parallel Methods
Schedule Author Index Session Index
Date: Wednesday, June 27Time: 10:30-12:15Location: MarkovChairman: Martin J. Gander, Felix Kwok and Yvon Maday
10:35-11:00 : Bankim Chandra MandalDirichlet-Neumann Waveform Relaxation for the Time Dependent HeatEquationAbstract
11:00-11:25 : Mohamed Kamel RiahiParareal in Time Intermediate Targets Methods for Optimal ControlProblemAbstract
11:25-11:50 : Ron HaynesA RIDC-DD Space-Time Algorithm for Time Dependent Partial DifferentialEquationsAbstract
11:50:12:15 : Olga Mula HernandezParareal for Neutronic Core CalculationsAbstract
M15 Abstract
Part 1
Part 2
194
Dirichlet-Neumann Waveform Relaxation for the TimeDependent Heat Equation
Session Schedule Author Index Session Index M15
Bankim Chandra MandalUniversity of [email protected]
Martin J. GanderUniversity of [email protected]
Felix KwokUniversity of [email protected]
AbstractWe present a waveform relaxation version of the Dirichlet-Neumann method forthe time dependent heat equation. Like the Dirichlet-Neumann method for steadyproblems, the method is based on a non-overlapping spatial domain decomposition,and the iteration involves subdomain solves with Dirichlet boundary conditionsfollowed by subdomain solves with Neumann boundary conditions. However, eachsubdomain problem is now in space and time, and the interface conditions are alsotime-dependent. An analysis using Laplace transforms shows linear convergencefor unbounded spatial domains, except for a very specific choice of the relaxationparameter, for which the method converges in a finite number of steps. A morerefined analysis on bounded domains reveals then that for this optimal choice ofthe relaxation parameter, we get superlinear convergence when we consider finitetime windows, similar to the case of Schwarz waveform relaxation algorithms. Theconvergence rate depends on the length of the subdomains as well as the size of thetime window. For any other choice of the relaxation parameter, convergence is onlylinear. We illustrate our theoretical results with numerical experiments.
195
Parareal in Time Intermediate Targets Methods for OptimalControl Problem
Session Schedule Author Index Session Index M15
Mohamed Kamel RiahiUPMC Univ Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions, Paris, [email protected]
Yvon MadayUPMC Univ Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions, Paris, FranceDivision of Applied Mathematics, Brown University, Providence, RI, [email protected]
Salomon JulienCEREMADE, Universite Paris-Dauphine, Pl. du Mal. de Lattre de Tassigny, F-75016, Paris, [email protected]
AbstractWe present a method that enables us to solve in parallel the Euler-Lagrange systemassociated with the optimal control of a parabolic equation. Our approach is basedon the definition and an iterative update of a sequence of intermediate targets thatgives rise to independent sub-problems that can be solved in parallel. In orderto accelerate the time-resolution, this method can be coupled with the pararealin time algorithm. Moreover, multi-dimensional optimization can be used withfull parallelization to achieve the best scalability. Numerical experiments show theefficiency of our methods.
196
A RIDC-DD Space-Time Algorithm for Time Dependent PartialDifferential Equations
Session Schedule Author Index Session Index M15
Ronald HaynesMemorial University of [email protected]
Benjamin OngMichigan State [email protected]
AbstractRecently, the Revisionist Integral Deferred Correction (RIDC) approach has beenshown to be a relatively easy way to add small scale parallelism (in time) to thesolution of time dependent PDEs. In this talk I will show how large scale spatialparallelism can be added to RIDC using relatively simple domain decompositionstrategies. This results in a truly parallel space-time method for PDEs suitable forhybrid OpenMP/MPI implementation. Initial scaling studies will demonstrate theviability of the approach.
197
Parareal for Neutronic Core Calculations
Session Schedule Author Index Session Index M15
Olga Mula HernandezCEA - Centre de Saclay DEN/DM2S/SERMA/[email protected]
Yvon MadayUPMC Univ Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions, Paris, FranceDivision of Applied Mathematics, Brown University, Providence, RI, [email protected]
Jean-Jacques LautardCEA - Centre de Saclay DEN/DM2S/SERMA/[email protected]
AbstractThe parareal in time algorithm is a time domain decomposition method for theapproximation of evolution problems. Its easy implementation in a parallel fashionallows for significant speed-ups in the computing time and opens the door to longtime computations that involve accurate propagators. In this talk, we first proposeto overview the different strategies for the parallelization of the algorithm. Thenwe will illustrate the efficiencies of these different strategies in a concrete PDE: thekinetic neutron diffusion equation in a nuclear reactor core. Implementations havebeen carried out with the MINOS solver, which is a tool developped at CEA inthe framework of the APOLLO3 R© project. As a conclusion, we will discuss thepossibility of using neutron diffusion as a coarse propagator for neutron transport.
198
Mini Symposium M16Domain Decomposition with MortarsSchedule Author Index Session Index
Organizers: Yvon Maday and Caroline Japhet
AbstractMortar methods are domain decomposition techniques based on a weak couplingbetween subdomains with nonconforming meshes, allowing different discretizationschemes or even different physical models on each sides of the non-conforming inter-face. Originally introduced for the coupling of finite element with spectral elementsmethods, these techniques are used in a large class of finite element discretizationsand for applications in computational electromagnetics, mechanics and fluid dynam-ics. There are still interesting theoretical and numerical difficulties in analysing apriori estimates for mortar coupling and in extending these methods to elastodynam-ics equations, to heterogeneous problems, to incompressible flows in axisymmetricchannels, to optimized Schwarz methods and to FETI methods. The aim of thisminisymposium is to report on recent advances in this field and on implementationissues with FreeFem++.
Part 1
Part 2
199
M16P1 Schedule
Domain Decomposition with Mortars
Schedule Author Index Session Index
Date: Monday, June 25Time: 10:30-12:15Location: AmphiChairman: Yvon Maday and Caroline Japhet
10:35-11:00 : Frederic HechtMortar Method to Solve Problem with Non-matching Grids in Freefem++Abstract
11:00-11:25 : Alfio QuarteroniDiscontinuous Approximation of Elastodynamics EquationsAbstract
11:25-11:50 : Caroline JaphetMortar Methods with Optimized Transmission ConditionsAbstract
11:50:12:15 : Zakaria BelhachmiSpectral Element Discretization of Incompressible Flows in AxisymmetricChannelsAbstract
M16 Abstract
Part 2
200
Mortar Method to Solve Problem with Non-matching Grids inFreefem++
Session Schedule Author Index Session Index M16
Frederic HechtUPMC Univ Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions, F-75005, Paris,[email protected]
Sylvain AuliacUPMC Univ Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions, F-75005, Paris,[email protected]
Faker Ben BelgacemLMAC, UTC, F-60205 Compiegne, [email protected]
AbstractFor a sake of simplicity, I will show how to solve classical physical problems withmultiple domains with non-matching grids. The examples are
1. A Poisson equation solved with a domain decomposition based on three fieldsmortar method.
2. Couplings between two thermal models with a Contact thermal (i.e F = [u]where F is the thermal flux, and [u] is the jump of the temperature).
3. A Signoriny interface problem (example of the IpOpt interface).
In these three examples, we show some mortar technics, and how they solve thisproblem numerically on a parallel computer with MPI interface and in two or threedimensions.
201
Discontinuous Approximation of Elastodynamics Equations
Session Schedule Author Index Session Index M16
Alfio QuarteroniMATHICSE-CMCS, EPFL, Lausanne (Switzerland) andMOX, Politecnico di Milano, Milan (Italy)[email protected]
AbstractThe possibility of inferring the physical parameter distribution of the Earth’s sub-stratum, from information provided by elastic wave propagations, has increased theinterest towards computational seismology. Recent developments have been focusedon spectral element methods. The reason relies on their flexibility in handling com-plex geometries, retaining the spatial exponential convergence for locally smoothsolutions and a natural high level of parallelism. In this talk, we consider a Dis-continuous Galerkin spectral element method (DGSEM) as well as discontinuousMortar methods (DMORTAR) to simulate seismic wave propagations in three di-mensional heterogeneous media. The main advantage with respect to conformingdiscretizations as those based on Spectral Element Method is that DG and Mor-tar discretizations can accommodate discontinuities, not only in the parameters,but also in the wavefield, while preserving the energy. The domain of interest Ωis assumed to be union of polygonal substructures Ωi. We allow this substructuredecomposition to be geometrically non-conforming. Inside each substructure Ωi,a conforming high order finite element space associated to a partition Thi
(Ωi) isintroduced. We allow the use of different polynomial approximation degrees withindifferent substructures. Applications to simulate benchmark problems as well asrealistic seismic wave propagation processes are presented.This work is in collaboration with P. Antonietti, I. Mazzieri and F. Rapetti.
202
Mortar Methods with Optimized Transmission Conditions
Session Schedule Author Index Session Index M16
Caroline JaphetLAGA, University Paris 13 and Project Pomdapi, INRIA, [email protected]
Yvon MadayUniversity Paris 6, Laboratoire Jacques-Louis Lions, Paris, FranceDivision of Applied Mathematics, Brown University, Providence, RI, [email protected]
Frederic NatafUniversity Paris 6, Laboratoire Jacques-Louis Lions, Paris, [email protected]
AbstractFor many applications in mechanics or fluid dynamics, one need to use differentdiscretizations in different regions of the computational domain to match with thephysical scales. Mortar methods are domain decomposition techniques based on aweak coupling between subdomains and enable the use of nonconforming grids. Onthe other hand, the optimized Schwarz methods, based on Robin or Ventcel trans-mission conditions greatly enhance the information exchange between subdomainsand lead to robust and fast algorithms. A new cement has been developed overthe last years which allows to glue nonconforming grids with Robin transmissionconditions for Schwarz type methods.We present this new cement for piecewise polynomials of low and high order in 2dand extended in 3d for P1 elements. The nonconforming domain decompositionmethod is proved to be well posed, and the error analysis is performed. Then wepresent numerical results that illustrate the method.
203
Spectral Element Discretization of Incompressible Flows inAxisymmetric Channels
Session Schedule Author Index Session Index M16
Zakaria BelhachmiLMIA, EA CNRS, Universite de Haute Alsace, Rue des Freres Lumiere, 68096 [email protected]
AbstractWe consider the Stokes and Navier-Stokes equations in a three-dimensional axisym-metric domain with non zero boundary conditions at the inward and outward faces.We propose a mortar spectral element discretization of this problem that relies onthe use of truncated Fourier series in the angular direction and spaces of polynomi-als on each element of a partition of the meridian domain. A similar discretizationin the framework of finite elements was considered in [1]. The key problem in theapplication of the method here is the choice of the quadrature formula: Indeed,the use of cylindrical coordinates leads to the appearance of a weight dependingon the radial variable. Moreover, since we intend to use curved elements to handlethe geometry of the initial domain, the discretization relies on the use of mappingswhich send a reference rectangle onto the curved element, so that we are led to con-sider problems with variable coefficients [2]. An idea to overcome this difficulty, dueto Y. Maday et E. Rnquist [3], consists in using over-integration, in the followingsense: The number of nodes in the quadrature formula is larger than the numberof degrees of freedom for each unknown. We perform the numerical analysis of thediscrete problem. Some numerical experiments for the Stokes system enable us tocheck the efficiency of the discretization.
[1] Z. Belhachmi, C. Bernardi, S. Deparis and F. Hecht, A truncated Fourier/finiteelement discretization of the Stokes equations in an axisymmetric domain, Math.Models Methods Appl. Sci., 16 (2006). 233–263.
[2] Z. Belhachmi, A. Karageorghis, Spectral element discretization of the Stokesequations in deformed axisymmetric geometries, Adv. Appl. Math. Mech. 3. 4(2011), 448-469.
[3] Y. Maday and E. M. Ronquist, Optimal error analysis of spectral methodswith emphasis on non-constants and deformed geometries, Comput. Methods Appl.Mech. Engrg., 80 (1990.), 91-115.
204
M16P2 Schedule
Domain Decomposition with Mortars
Schedule Author Index Session Index
Date: Monday, June 25Time: 16:00-18:10Location: AmphiChairman: Yvon Maday and Caroline Japhet
16:05-16:30 : Christian WalugaQuasi-Optimal a priori Estimates for the Lagrange Multiplier in MortarType CouplingsAbstract
16:30-16:55 : Yvon MadaySome Recent Applications of Non Conforming ApproximationsAbstract
16:55-17:20 : Francois-Xavier RouxFETI-2LM for Localizing the MortarsAbstract
17:20-17:45 : Oldrich VlachOn Effective Implementation of the Non-penetration Condition forNon-matching Grids Preserving Scalability of FETI Based AlgorithmsAbstract
17:45-18:10 : Todd ArbogastMultiscale Mortar Mixed Methods for Heterogeneous Elliptic ProblemsAbstract
M16 Abstract
Part 1
205
Quasi-optimal A Priori Estimates for the Lagrange Multiplier inMortar Type Couplings
Session Schedule Author Index Session Index M16
Christian WalugaM2 Zentrum Mathematik, Technische Universitat [email protected]
Jens Markus MelenkInstitut fur Analysis und Scientific Computing, Technische Universitat [email protected]
Barbara WohlmuthM2 Zentrum Mathematik, Technische Universitat [email protected]
AbstractWe present quasi-optimal a priori convergence results for the approximation of sur-face based Lagrange multipliers such as those employed in mortar type finite elementcouplings. In classical estimates based on the standard saddle point theory, the er-ror estimates for both the primal and dual variables are obtained simultaneously,which results in suboptimal a priori estimates for the dual variable. While improvedestimates are fairly easily achievable if optimal order L∞ bounds are available, thepresent analysis is based on significantly weaker regularity requirements. By usingnew estimates for the primal variable in strips of width O(h) near the mortar inter-faces, we illustrate that an additional factor
√h| lnh| in the a priori bound for the
dual variable can be recovered, where the logarithmic factor can even be droppedfor higher order elements (p > 1). We outline the analysis of a second order ellip-tic model problem and discuss possible extensions of the theory. Finally, we givedifferent numerical examples to support the theoretical results.
206
Some Recent Applications of Non Conforming Approximations
Session Schedule Author Index Session Index M16
Yvon MadayUPMC Univ Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions, F-75005, Paris,FranceDivision of Applied Mathematics, Brown University, Providence, RI, [email protected]
AbstractThe original mortar element method has been introduced in order to be able tocouple different type of variational discretizations used on different non overlap-ping subdomains. The matching on the interface between two subdomains is donethrough a Lagrange multiplier and the mortar method specifies an easy way to de-fine these multipliers in order that the approximation is optimal in terms of degreesof freedom.We present here some new applications of the mortar technology in the differentframework of reduced basis element approximation for parameter dependent PDE’s.The solution to such problems over each subdomains is approximated by a linearcombination of well chosen solutions of a similar problem on the considered subdo-main, the different approximations need to be glued at the interface with mortartype approaches. The approach can be used in conjunction with data assimilationwhere in some subdomains, the approximation is constructed in order, not to satisfythe PDE but match some measured data.
207
FETI-2LM for Localizing the Mortars
Session Schedule Author Index Session Index M16
Francois-Xavier [email protected]
AbstractNon conforming interfaces are generally introduced for modelling purpose, in spe-cific areas. In such a context, there are only a few subdomains that are generallynot balanced. The design of efficient parallel solution methods using domain de-composition requires a second level splitting that may create crosspoints on thenon conforming interfaces where the mortar conditions couple more than two sub-domains in a complex way. In order to avoid this drawback, all interface nodesassociated with one mortar on each side of the initial non conforming interfacemust be attached to only one subdomain. Attaching extra nodes to a subdomainleads to ill posed local problems. With the FETI-2LM method this problem canbe handled thanks to the Robin interface conditions. In this paper, we presentthe technique for relocalizing mortar conditions with the FETI-2LM method in thecase of multi-level splitting. Results of numerical experiments will illustrate thepresentation.
208
On Effective Implementation of the Non-penetration Conditionfor Non-matching Grids Preserving Scalability of FETI BasedAlgorithms
Session Schedule Author Index Session Index M16
Oldrich VlachIT4Innovations, VSB-Technical University of Ostrava, 17. listopadu 15/2172, 708 33Ostrava Poruba, Czech [email protected]
Zdenek DostalIT4Innovations, VSB-Technical University of Ostrava, 17. listopadu 15/2172, 708 33Ostrava Poruba, Czech [email protected]
Tomas BrzobohatyIT4Innovations, VSB-Technical University of Ostrava, 17. listopadu 15/2172, 708 33Ostrava Poruba, Czech [email protected]
AbstractMathematical models of contact include the inequalities which make the contactproblems strongly nonlinear. In spite of this, a number of interesting results havebeen obtained by modifications of the methods that were known to be scalable forlinear problems, in particular of the FETI domain decomposition method introducedby Farhat and Roux for parallel solution of linear problems. The point of thispaper is to extend our results obtained for elastic contact problems to the contactproblems with non-matching grids which necessarily emerge, e.g., in the solutionof transient contact problems or in contact shape optimization. We want to getgood approximation and the constraint matrix B with nearly orthogonal rows. Weconsider both standard engineering approaches such as node to segment, or mortarelements. We give simple bounds on the singular values of the resulting matrix Band results of numerical experiments, including both the academic examples andsome problems of practical interest. We conclude that the normalized orthogonalmortars proposed by Wohlmuth can be used to approximate the non-penetrationconditions in a way that complies with the requirements of the FETI methods.
209
Multiscale Mortar Mixed Methods for Heterogeneous EllipticProblems
Session Schedule Author Index Session Index M16
Todd ArbogastUniversity of Texas, Austin, TX, [email protected]
Hailong XiaoUniversity of Texas, Austin, TX, [email protected]
AbstractWe consider the problem of computing flow fields in porous media with extreme nat-ural heterogeneities. The system is modeled by a second order elliptic problem witha heterogeneous coefficient, which we write in mixed form. We develop numericalapproximations suitable for parallel computation through the use of nonoverlappingdomain decomposition mortar methods with a restricted set of degrees of freedomon the interfaces. We devise an effective but purely local multiscale method thatincorporates information from homogenization theory. In the case of a locally peri-odic heterogeneous coefficient of period ε, we prove that the new method achievesboth optimal order error estimates in the discretization parameters and conver-gence when ε is small. We also use this mortar approach to devise preconditionersthat incorporate exact coarse-scale information to iteratively solve the full fine-scaleproblem. Moreover, we present numerical examples to assess the performance ofthe techniques.
210
Mini Symposium M17Domain Decomposition Methods based on RobinConditions for Large and / or Nonlinear Prob-lemsSchedule Author Index Session Index
Organizers: Heiko Berninger, Sebastien Loisel, Oliver Sander
AbstractDomain decomposition methods can be used to solve a wide variety of problems.In parallelized nonoverlapping settings, at each iteration, processors exchange dataalong an interface that separates the subdomains. In order to give improved con-vergence properties, one can choose optimized transmission conditions across theinterface. These optimized transmission conditions should be chosen in such a waythat the subdomain problems can be solved readily while maintaining fast conver-gence. In recent years, Robin boundary conditions have been used and optimizedin a variety of linear and nonlinear problems, and some large-scale implementationsare being developed. In this minisymposium, we will discuss recent developmentsin domain decomposition methods with Robin transmission conditions as well astheir optimizations.
Part 1
Part 2
211
M17P1 Schedule
Domain Decomposition Methods based on Robin Con-ditions for Large and / or Nonlinear Problems
Schedule Author Index Session Index
Date: Thursday, June 28Time: 10:30-12:15Location: AmphiChairman: Heiko Berninger, Sebastien Loisel, Oliver Sander
10:35-11:00 : Sebastien LoiselLarge-Scale Implementation of Optimized Decomposition MethodsAbstract
11:00-11:25 : Florence HubertOptimized Schwarz Algorithms for Anisotropic Elliptic Operators in theFramework of DDFV SchemesAbstract
11:25-11:50 : Oliver SanderThe 2-Lagrange-Multiplier Method for the Richards EquationAbstract
11:50:12:15 : Minh Binh TranOptimized Schwarz Methods for the Primitive EquationsAbstract
M17 Abstract
Part 2
212
Large-Scale Implementation of Optimized DecompositionMethods
Session Schedule Author Index Session Index M17
Sebastien LoiselHeriot-Watt [email protected]
Anastasios KarangelisHeriot-Watt [email protected]
Chris MaynardUniversity of [email protected]
AbstractDomain decomposition methods are used to solve large elliptic problems in parallel.The 2-Lagrange multiplier method is a domain decomposition method which usesoptimized Robin boundary conditions on the artificial interface and which is relatedto the optimized Schwarz method. In this talk, we will describe a massively parallelimplementation of the 2-Lagrange multiplier method which runs on the Hectorsupercomputer.
213
Optimized Schwarz Algorithms for Anisotropic Elliptic Operatorsin the Framework of DDFV Schemes
Session Schedule Author Index Session Index M17
Florence HubertLATP-Aix-Marseille [email protected]
Martin J. GanderUniversity of [email protected]
Stella KrellLaboratoire Dieudonne, Universite de Nice - Sophia [email protected]
AbstractClassical and optimized Schwarz algorithms have been developped for anisotropicelliptic problems using Discrete Duality Finite Volume techniques (DDFV) over thelast five years. Like for Discontinuous Galerkin method (DG), it is not a priori clearhow to appropriately discretize transmission conditions in DDFV, and numericalexperiments have shown that very different scalings both for the optimized param-eters and the contraction rates of the algorithms can be obtained, depending on thediscretization. We explain in this presentation how the DDFV discretization caninfluence the performance of Schwarz algorithms, and also propose a new DDFVdiscretization technique of interface conditions which leads to the expected conver-gence rate of the Schwarz algorithms obtained from an analysis at the continuouslevel.
214
The 2-Lagrange-Multiplier Method for the Richards Equation
Session Schedule Author Index Session Index M17
Oliver SanderFreie Universitat [email protected]
Heiko BerningerUniversite de [email protected]
Sebastien LoiselHeriot-Watt [email protected]
AbstractThe 2-Lagrange-multiplier method was originally introduced to solve linear ellipticequations on domains with a nonoverlapping partition with cross-points. It worksby rewriting the problem as a set of local Robin boundary problems, and iterat-ing on the Robin boundary data. In this talk we generalize the method to theRichards equation for unsaturated porous media flow. As in the linear case, localRobin problems have to be solved. By applying the Kirchhoff transformation toeach subdomain problem we obtain a set of strictly convex minimization problems,for which a fast and robust multigrid solver is available. This works even if thepermeability and saturation functions are different on each subdomain. Therefore,the 2-Lagrange-multiplier method extends a previous approach of Berninger, Korn-huber, and Sander for layered soils to general decompositions with cross-points.
215
Optimized Schwarz Methods for the Primitive Equations
Session Schedule Author Index Session Index M17
Minh-Binh TranBCAM, Basque Center for Applied [email protected]
Laurence HalpernLAGA, Universite Paris [email protected]
AbstractThe primitive equations are equations, describing large scale dynamics of oceans andatmosphere. These equation are derived from the Navier-Stokes equations, with ro-tation, coupled to thermodynamics and salinity diffusion-transport equations, whichaccount for the buoyancy forces and stratification effects under the Boussinesq ap-proximation. In this talk, we will give numerical and theoretical results for Schwarzdomain decomposition methods with Robin transmission conditions for this equa-tion.
216
M17P2 Schedule
Domain Decomposition Methods based on Robin Con-ditions for Large and / or Nonlinear Problems
Schedule Author Index Session Index
Date: Friday, June 29Time: 10:30-12:15Location: AmphiChairman: Heiko Berninger, Sebastien Loisel, Oliver Sander
10:35-11:00 : Soheil HajianDiscontinuous Galerkin, Block Jacobi and Schwarz MethodsAbstract
11:00-11:25 : Ronald HaynesAn Optimized Schwarz Method for the Generation of Equidistributed GridsAbstract
11:25-11:50 : Joel PhillipsSchwarz Methods for Plane Wave Discontinuous Galerkin MethodsAbstract
11:50:12:15 : Yingxiang XuThe Influence of Interface Curvature on Transmission Conditions in DomainDecomposition MethodsAbstract
M17 Abstract
Part 1
217
Discontinuous Galerkin, Block Jacobi and Schwarz Methods
Session Schedule Author Index Session Index M17
Soheil HajianUniversity of [email protected]
Martin J. GanderUniversity of [email protected]
AbstractFor classical discretizations of elliptic partial differential equations, like conformingfinite element methods (FEM) or finite difference methods (FDM), block Jacobiiterations are equivalent to classical Schwarz iterations with Dirichlet transmissionconditions. This is however not necessarily the case for discontinuous Galerkinfinite element methods (DGFEM). We will show in this talk for the model problem(η − ∆)u = f and various DGFEM discretizations that a block Jacobi methodapplied to the discretized problem can be interpreted as a Schwarz method withdifferent transmission conditions from the classical Dirichlet ones. We illustrate ourresults with numerical experiments.
218
An Optimized Schwarz Method for the Generation ofEquidistributed Grids
Session Schedule Author Index Session Index M17
Ronald HaynesMemorial University of [email protected]
AbstractAdaptively choosing an underlying grid for computation has proven to be a useful,if not essential, tool for the solution of boundary value problems and partial differ-ential equations. One way of generating adaptive meshes is through the so-calledequidistribution principle (EP). In one spatial dimension the required mesh can beobtained through the solution of a nonlinear BVP. In this talk I will review theidea of EP and consider the solution of the resulting BVP via an optimized Schwarziteration. Recent work on the generation of 2D locally equidistributed grids will bepresented.
219
Schwarz Methods for Plane Wave Discontinuous GalerkinMethods
Session Schedule Author Index Session Index M17
Joel PhillipsUniversity College, [email protected]
Timo BetckeUniversity College, [email protected]
Martin J. GanderUniversite de [email protected]
AbstractThe Ultra-Weak Variational Formulation (UWVF) and Plane Wave DiscontinuousGalerkin (PWDG) methods use bases composed of element-wise particular solu-tions. For Helmholtz’ problem, the weak form is∫
∂K
uh · ∇v · ndS − ik∫∂K
σh · nvdS =
∫K
fvdV,
where the fluxes, uh and σh are given in terms of jumps and averages. For example,on interior faces, we take
uh = uh+ τ · [[uh]]− β
ik[[∇huh]].
These methods give accurate approximations with small numbers of degrees of free-dom but also suffer from poor conditioning. In this talk, we will demonstrate howthis can be improved using a Schwarz-type domain decomposition.We will also show how a (purely algebraic) block-Jacobi relaxation of the DG formu-lation is equivalent to a non-overlapping Schwarz method with Robin transmissionconditions of the form,
∂u(n+1)1
∂n1+ su
(n+1)1 =
∂u(n)2
∂n1+ su
(n)2 ,
with s = ik, and how, with a little extra work, we can let s take more generalvalues.
220
The Influence of Interface Curvature on Transmission Conditionsin Domain Decomposition Methods
Session Schedule Author Index Session Index M17
Yingxiang Xu1. School of Mathematics and Statistics, Northeast Normal University2. Section de Mathematiques, Universite [email protected]
Martin J. GanderSection de Mathematiques, Universite de [email protected]
AbstractThe interface curvature, generally given by the concrete problem to be solved anddomain decomposition used, may affect greatly the convergence properties for do-main decomposition methods. We show in this talk for the Schwarz algorithmapplied to symmetric positive definite problems several transmission conditions, de-rived from both micro-local analysis and optimization of the convergence factorson a model problem. We observe that many of our transmission conditions in-volve the interface curvature parameter, and this leads to better performance of theassociated method generally. Overlap permits to accelerate the method, and theoptimization-based methods achieve the best performance. Our theoretical analysisand our numerical experiments show that for getting the best possible performance,the interface curvature should definitely be taken into account.
221
222
Mini Symposium M18Solvers for Discontinuous Galerkin MethodsSchedule Author Index Session Index
Organizers: Blanca Ayuso de Dios, Susanne C. Brenner
AbstractDiscontinuous Galerkin (DG) finite element methods were introduced in the late1970s, and they have undergone a rapid development in recent years. For this fam-ily of numerical techniques, the finite element spaces are not subject to inter-elementcontinuity conditions and local element spaces can be defined independently fromeach other. They possess many advantageous properties (local conservation; flexibil-ity in handling irregular meshes with hanging nodes and in designing hp-refinementstrategies; built-in parallelism) which have rendered them suitable for the approx-imation of a wide variety of problems; including elliptic, first-order hyperbolic andunsteady problems. They also provide a far-reaching framework to develop new ap-propriate discretization schemes for several applications where classical approachesmight fail.
Despite the versatility of DG techniques, their practical use has been oftenlimited by the much larger number of degrees of-freedom it requires compared withother classical discretization methods. For this reason, the development of effi-cient solvers and preconditioning strategies for the solution of the algebraic systemsresulting from DG discretizations is becoming crucial. Over the last ten years, dif-ferent solvers and preconditioning strategies based on domain decomposition (DD),multigrid and multilevel methods have been developed and analyzed for DG dis-cretizations of several (still simple) problems. Their complete understanding ishowever still lacking.
The aim of this mini-symposium is to bring together experts in the field todiscuss and identify the most relevant aspects of the current development of solu-tion techniques for DG methods. Sample topics include the design, the theoreticalanalysis and issues related to the implementation and applications of the varioussolution techniques.
Part 1
Part 2
223
M18P1 Schedule
Solvers for Discontinuous Galerkin Methods
Schedule Author Index Session Index
Date: Wednesday, June 27Time: 16:00-17:45Location: MarkovChairman: Blanca Ayuso de Dios, Susanne C. Brenner
16:05-16:30 : Alexandre PieriBDDC Preconditioners: from hp-Continuous to Discontinuous GalerkinSchemes with Different Local Polynomial DegreesAbstract
16:30-16:55 : Kolja BrixRobust Preconditioners for DG-Discretizations with Arbitrary PolynomialDegrees on Locally Refined MeshesAbstract
16:55-17:20 : Christoph LehrenfeldDD Preconditioning for High Order Hybrid DG Methods on TetrahedralMeshesAbstract
17:20-17:45 : Eun-Hee ParkA BDDC Method for a Symmetric Interior Penalty Galerkin MethodAbstract
M18 Abstract
Part 2
224
BDDC Preconditioners: from hp-Continuous to DiscontinuousGalerkin Schemes with Different Local Polynomial Degrees
Session Schedule Author Index Session Index M18
Alexandre B. PieriEcole Centrale de Lyon, [email protected]
Luca Franco PavarinoUniversity of Milan, [email protected]
Claudio CanutoPolitecnico di Torino, [email protected]
AbstractSpectral element methods can achieve exponential convergence rate for elliptic prob-lems and can be associated with domain decomposition algorithms to further achievescalable parallel solvers. The regularity of the solution may vary locally and adap-tive meshes with locally varying spectral discretization can improve accuracy. Here,we first extend the classic spectral element method with uniform polynomial degreesto a spectral formulation allowing local changes in polynomial degrees (relative toa spectral element). Then, we propose an efficient preconditioner to solve the asso-ciated algebraic system, extending the so-called Balancing Domain Decompositionby Constraints (BDDC) method developed for finite and spectral elements withuniform polynomial degrees to the case with jumps in polynomial degrees betweenelements. We also study a Discontinuous Galerkin formulation based on the Aux-iliary Space Method and we extend the BDDC preconditioner to this formulation.The results of several numerical tests in two dimensions for both continuous anddiscontinuous Galerkin formulations show the scalability, quasi-optimality and effi-ciency of the proposed method, as well as its robustness with respect to jumps inthe elliptic coefficients and local spectral degrees.
225
Robust Preconditioners for DG-Discretizations with ArbitraryPolynomial Degrees on Locally Refined Meshes
Session Schedule Author Index Session Index M18
Kolja BrixRWTH [email protected]
Martin Campos PintoIRMA Strasbourg - CNRS & Universite Louis [email protected]
Claudio CanutoPolitecnico di [email protected]
Wolfgang DahmenRWTH [email protected]
AbstractDiscontinuous Galerkin (DG) methods offer enormous flexibility regarding localgrid refinement and variation of polynomial degrees rendering such concepts pow-erful discretization tools. At the same time they have proven to be well-suitedfor a variety of different problem classes. While initially the main focus has beenon transport problems like hyperbolic conservation laws, interest has meanwhileshifted towards diffusion problems. We therefore consider DG discretizations forelliptic boundary value problems and in particular focus on the efficient solution ofthe linear systems of equations that arise from the Symmetric Interior Penalty DGmethod. We propose preconditioners that are based on the concept of the auxil-iary space method in combination with techniques from spectral element methodssuch as Legendre-Gauss-Lobatto grids. Under mild grading conditions on the gridrefinement and the variation of the polynomial degrees, we can show the resultingcondition numbers to be bounded even for locally refined grids with hanging nodesand arbitrary polynomial degrees. Special measures have to be taken in the caseof varying polynomial degrees around a hanging node. We present some numericalexperiments that demonstrate the efficiency of the preconditioners.
226
DD Preconditioning for High Order Hybrid DG Methods onTetrahedral Meshes
Session Schedule Author Index Session Index M18
Christoph LehrenfeldIGPM, RWTH [email protected]
Joachim SchoberlCME, TU [email protected]
AbstractDiscontinuous Galerkin methods are popular discretization methods in applica-tions from fluid dynamics and many others. The concept of hybridization offersa structure to reduce the discrete system to unknowns on the element interfaces [1]which makes arising linear systems more suitable for an efficient solution. Afterthe presentation of the method, we show for an elliptic model equation discretizedon tetrahedral meshes how standard domain decomposition techniques like non-overlapping Schwarz type methods or balancing domain decomposition with con-straints (BDDC) [2,3] can be easily applied to the Hybrid Discontinuous Galerkinformulation. Therefore we consider one element as a sub-domain and divide thedegrees of freedom into mean values on faces and the remainder as primal and dualunknowns. We prove poly-logarithmic (in the polynomial order) condition numberestimates for the preconditioned matrix. In order to show that, we have to developtechnical tools, specially that an optimal extension from faces to elements withDirichlet constraints are nearly as good as an extension without constraints.
[1] B. Cockburn, J. Gopalakrishnan, and R. Lazarov. Unified hybridization of dis-continuous Galerkin, mixed, and continuous Galerkin methods for second orderelliptic problems. SIAM J. Numer. Anal., 47:1319-1365, 2009.
[2] C. R. Dohrmann. A preconditioner for substructuring based on constrained en-ergy minimization. SIAM J. Sci. Comput., 25(1):246-258, 2003
[3] M. Dryja and O. B. Widlund. Towards a unified theory of domain decomposi-tion algorithms for elliptic problems. In T. F. Chan, R. Glowinski, J. Periaux, andO. B. Widlund, editors, Third International Symposium on Domain DecompositionMethods for Partial Differential Equations, pages 3–21, Philadelphia, 1990. SIAM.
227
A BDDC Preconditioner for a Symmetric Interior PenaltyGalerkin Method
Session Schedule Author Index Session Index M18
Eun-Hee ParkLouisiana State [email protected]
Susanne C. BrennerLouisiana State [email protected]
Li-yeng SungLouisiana State [email protected]
AbstractIn this talk we will discuss a nonoverlapping domain decomposition preconditionerfor a symmetric interior penalty Galerkin method for the heterogeneous ellipticproblem. The preconditioner is based on balancing domain decomposition by con-straints (BDDC). Theoretical results on the condition number estimate of the pre-conditioned system will be presented along with numerical results.
228
M18P2 Schedule
Solvers for Discontinuous Galerkin Methods
Schedule Author Index Session Index
Date: Thursday, June 28Time: 10:30-12:15Location: MarkovChairman: Blanca Ayuso de Dios, Susanne C. Brenner
10:35-11:00 : Paola F. AntoniettiSchwarz Methods for a Preconditioned WOPSIP Discretization of EllipticProblemsAbstract
11:00-11:25 : Andrew BarkerAdditive Schwarz Preconditioners for the Discontinuous Petrov-GalerkinMethodAbstract
11:25-11:50 : Guido KanschatMultigrid Methods for a Divergence–Conforming DG Discretization ofIncompressible FlowAbstract
11:50:12:15 : Ludmil T. ZikatanovA Preconditioner for H(div)-Conforming DG Discretizations of StokesEquationAbstract
M18 Abstract
Part 1
229
Schwarz Methods for a Preconditioned WOPSIP Discretizationof Elliptic Problems
Session Schedule Author Index Session Index M18
Paola F. AntoniettiMOX, Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo daVinci 32, 20133 Milano, [email protected]
Blanca Ayuso de DiosCentre de Recerca Matematica, Campus de Bellaterra, 08193 Bellaterra (Barcelona),[email protected]
Susanne C. BrennerDepartment of Mathematics and Center for Computation & Technology, LouisianaState University, Baton Rouge, LA 70803, [email protected]
Li-yeng SungDepartment of Mathematics and Center for Computation & Technology, LouisianaState University, Baton Rouge, LA 70803, [email protected]
AbstractWe propose and analyze several two-level non-overlapping Schwarz methods for apreconditioned weakly over-penalized symmetric interior penalty (WOPSIP) dis-cretization of a second order boundary value problem. We show that the conditionnumber of the resulting preconditioned linear systems of equations is of order H/h,being H and h the granularity of the coarse and fine partitions, respectively. Numer-ical experiments that illustrate the performance of the proposed two-level Schwarzmethods are also presented.
230
Additive Schwarz Preconditioners for the DiscontinuousPetrov-Galerkin Method
Session Schedule Author Index Session Index M18
Andrew T. BarkerLouisiana State [email protected]
Susanne C. BrennerLouisiana State [email protected]
Eun-Hee ParkLouisiana State [email protected]
Li-yeng SungLouisiana State [email protected]
AbstractThe discontinuous Petrov-Galerkin method allows for use of nearly optimal testfunctions at a reasonable computational cost, because the test functions can besolved for locally. The resulting methods can be very effective and show good sta-bility properties, but solution of the resulting ill-conditioned linear systems is achallenge. We explore the effectiveness of additive Schwarz preconditioning for lin-ear systems arising from the DPG discretization, considering both their theoreticalproperties and their practical efficiency.
231
Multigrid Methods for a Divergence-Conforming DGDiscretization of Incompressible Flow
Session Schedule Author Index Session Index M18
Guido KanschatTexas A&M [email protected]
Youli MaoTexas A&M [email protected]
AbstractA multigrid method based on an overlapping domain decomposition smoother ispresented. The smoother operates in the divergence free subspace and thus doesnot require to be embedded into a block preconditioner for the saddle point problem.Its efficiency is documented with numerical examples.
232
A Preconditioner for H(div)-Conforming DG Discretizations ofStokes Equation
Session Schedule Author Index Session Index M18
Ludmil ZikatanovDepartment of Mathematics, The Pennsylvania State University, University Park,PA 16802, [email protected]
Blanca Ayuso de DiosCentre de Recerca Matematica, UAB Science Faculty, 08193 Bellaterra, Barcelona,[email protected]
Franco BrezziIMATI del CNR and Istituto Universitario di Studi Superiori (IUSS), 27100 Pavia,[email protected]
L. Donatella MariniDipartimento di Matematica, Universita degli Studi di Pavia and IMATI del CNR,27100 Pavia, [email protected]
Jinchao XuDepartment of Mathematics, The Pennsylvania State University, University Park,PA 16802, [email protected]
AbstractWe present a preconditioner forH(div) conforming, DG-discretizations of the Stokesequation. We focus on preconditioning the linear system resulting from the lowestorder Brezzi-Douglas-Marini (BDM) elements discretization for the velocity. Wesolve the problem on the divergence free subspace using an auxiliary space precon-ditioner. The action of the preconditioner amounts to solving two scalar Laplaceequations and a vector Laplace equation. We present numerical tests that show thatsuch preconditioner is optimal and we also discuss some of the theoretical estimateson the condition number of the preconditioned system.
233
234
Mini Symposium M19Domain Decomposition in Computational Car-diologySchedule Author Index Session Index
Organizers: Rolf Krause and Luca Pavarino
AbstractThe numerical simulation of the mechanical and electrical activity within the humanheart is a challenging task. Different effects as the propagation of the elecricalactivation front, the chemical reactions within the ion channels, or the influence ofthe fiber orientation on the mechanical contraction of the heart have to be taken intoacount. High spatial and temporal resolution is necessary in order to resolve, e.g.,electrophysiological phenomena as the activation front, or mechanicaly importantquantities as the micro-structure of the tissue. As a consequence, the employedsimulation methods have to be adapted carefully. This includes possibly adaptivediscretization methods as well as efficient parallel solution techniques for the arisinglarge scale problems. This minisymposium is intended to provide a platform topresent and to discuss newest developments in computational cardiology with astrong focus on domain decomposition and multigrid techniques.
Part 1
Part 2
235
M19P1 Schedule
Domain Decomposition in Computational Cardiology
Schedule Author Index Session Index
Date: Thursday, June 28Time: 10:30-12:15Location: PetriChairman: Rolf Krause and Luca Pavarino
10:35-11:00 : Luca Gerardo-GiordaOptimized Schwarz Coupling and Model Adaptivity for NumericalElctrocardiologyAbstract
11:00-11:25 : Dorian KrauseScalable Solvers for Electrocardiology on Massively Parallel ArchitecturesAbstract
11:25-11:50 : Stefano ZampiniExact and Inexact BDDC Methods for the Cardiac Bidomain ModelAbstract
11:50-12:15 : Charles PierreA Preconditioner with Almost Linear Complexity for the Bidomain ModelAbstract
M19 Abstract
Part 2
236
Optimized Schwarz Coupling and Model Adaptivity forNumerical Elctrocardiology
Session Schedule Author Index Session Index M19
Luca Gerardo-GiordaBCAM - Basque Center for Applied Mathematics, Bilbao, [email protected]
Lucia MirabellaGeorgia Tech, Atlanta, [email protected]
Mauro PeregoFlorida State University, Thallahassee, [email protected]
Alessandro VenezianiEmory University, Atlanta, [email protected]
AbstractThe Bidomain model is nowadays one of the most accurate mathematical descrip-tions of the action potential propagation in the heart. However, its numericalapproximation is in general fairly expensive as a consequence of the mathemati-cal features of this system. For this reason, a simplification of this model, calledMonodomain problem is often adopted in order to reduce computational costs. Re-liability of this model is questionable in the presence of applied currents and in theregions where the upstroke or the late recovery of the action potential is occurring,but in the absence of applied currents it provides a reasonable approximation for theaction potential propagation at the heart scale. An heterogeneous approach aimingat reducing computational costs and maintaining accuracy can be devised by solvingthe Bidomain problem only over “critical” regions of the domain (the term“critical”being driven by physiopathological arguments), and solving the Monodomain prob-lem in areas where the potential propagation dynamics does not require the mostsophisticated model. This approach falls in the general framework of “model adap-tivity”.In this talk, stemming from an intermediate model called Hybridomain, we willdescribe a model adaptive strategy: the computational domain is subdivided intoregions, coupled through an Optimized Schwarz Method, in which either the Bido-main or the Monodomain problem is solved. The model choice is driven by a modelerror estimator following the spatio-temporal evolution of the action potential prop-agation.
237
Scalable Solvers for Electrocardiology On Massively ParallelArchitectures
Session Schedule Author Index Session Index M19
Dorian KrauseInstitute of Computational Science, Universita della Svizzera italiana,Via Giuseppe Buffi 13, 6904 Lugano, [email protected]
Mark PotseInstitute of Computational Science, Universita della Svizzera italiana,Via Giuseppe Buffi 13, 6904 Lugano, [email protected]
Thomas DickopfInstitute of Computational Science, Universita della Svizzera italiana,Via Giuseppe Buffi 13, 6904 Lugano, [email protected]
Rolf KrauseInstitute of Computational Science, Universita della Svizzera italiana,Via Giuseppe Buffi 13, 6904 Lugano, [email protected]
AbstractThe accurate simulation of the electrical propagation in cardiac tissue is a challeng-ing problem as a large span of length- and timescales needs to be resolved. Thisis even more the case when dealing with full heart simulations on complicated ge-ometries obtained from medical imaging data. High performance computing and –in particular – massively parallel processing are important enabling techniques forcomputational electrocardiology. In this talk we describe algorithms and implemen-tation techniques for exploiting contemporary massively parallel architectures forsolving mono- and bidomain reaction diffusion equations. We present the hybridparallelization of a heart model and analyze the advantages of multithreading forthis application for explicit and implicit-explicit time integration on up to 8,448cores of a Cray XT5 system. Strong and weak scalability of several precondition-ers (ranging from Block-Jacobi ILU(0) to algebraic multigrid) for the linear systemarising in the parabolic-elliptic formulation of the bidomain equation are investi-gated. We conclude the presentation by discussing the results of a recent study ofthe accuracy of signals computed at low resolution using sources from high reso-lution propagation simulations. This application showcases the capabilities of ourapproach on massively parallel architectures and highlights the gains achievable bycarefully optimizing the mesh resolution used at the different stages of our simula-tion workflow.
238
Exact and Inexact BDDC Methods for the Cardiac BidomainModel
Session Schedule Author Index Session Index M19
Stefano ZampiniCASPUR, Via dei Tizii 6, 00185 [email protected]
AbstractBalancing Domain Decomposition by Constraints (BDDC) preconditioners are con-structed and analyzed for an implicit-explicit discretization of the parabolic-parabolicformulation of the cardiac Bidomain model, discretized with low-order conformingisoparametric Q1 elements in three dimensions. Unlike more conventional non-overlapping methods, the preconditioner’s construction is carried out on the wholeset of dofs allowing for the use of approximate solvers on the subdomains. Afterhaving introduced the formulation of the Bidomain variational problem and pre-conditioner details, theoretical estimates will be provided for the upper bound ofthe average operator in the Bidomain Schur norm; parallel experimental results willconfirm quasi-optimality and scalability of the exact BDDC method for the cardiacBidomain model in three-dimensions. Theoretical and experimental results will alsoconfirm robustness of the BDDC method in case of jumps in cardiac conductivitycoefficients aligned with the interface of the non-overlaping partition. Finally, sincememory consumption is the most important bottleneck of all non-overlapping meth-ods in three dimensions, an inexact BDDC formulation is considered, substitutingexact factorizations of local problems by the action of algebraic multigrid precon-ditioners: different types of smoothers will be also taken into account. Robustnessand scalability of the approximate preconditioner will be theoretically proved andexperimentally validated; experimental results for large simulations up to 60 millionof unknowns will confirm the efficiency of the inexact approach.
239
A Preconditioner with Almost Linear Complexity for theBidomain Model
Session Schedule Author Index Session Index M19
Charles PierreLMA, Universite de Pau et des pays de l’[email protected]
AbstractThe bidomain model is widely used in electro-cardiology to simulate spreading ofexcitation in the myocardium and electrocardiograms. It consists of a system oftwo parabolic reaction diffusion equations coupled with an ODE system. Its dis-cretisation displays an ill-conditioned system matrix to be inverted at each timestep: simulations based on the bidomain model therefore are associated with highcomputational costs. In this paper we propose a preconditioning for the bidomainmodel in an extended framework including a coupling with the surrounding tissues(the torso). The preconditioning is based on a formulation of the discrete problemthat is shown to be symmetric positive semi-definite. A block LU decomposition ofthe system together with a heuristic approximation (referred to as the monodomainapproximation) are the key ingredients for the preconditioning definition. Numeri-cal results are provided for two test cases: a 2D test case on a realistic slice of thethorax based on a segmented heart medical image geometry, a 3D test case involv-ing a small cubic slab of tissue with orthotropic anisotropy. The analysis of theresulting computational cost (both in terms of CPU time and of iteration number)shows an almost linear complexity with the problem size, i.e. of type n logα(n) (forsome constant α) which is optimal complexity for such problems.
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M19P2 Schedule
Domain Decomposition in Computational Cardiology
Schedule Author Index Session Index
Date: Friday, June 29Time: 10:30-12:15Location: PetriChairman: Rolf Krause and Luca Pavarino
10:35-11:00 : Martin WeiserDelayed Residual Compensation for Bidomain EquationsAbstract
11:00-11:25 : Gernot PlankGPU Accelerated Strongly Scalable Simulations of Cardiac electro-mechanicsAbstract
11:25-11:50 : Ricardo Ruiz BaierAn Eulerian Finite Element Method for the Simulation of CardiomyocyteActive ContractionAbstract
11:50:12:15 : Maxime SermesantInteractive Electromechanical Model of the Heart for Patient-SpecificSimulationAbstract
M19 Abstract
Part 1
241
Delayed Residual Compensation for Bidomain Equations
Session Schedule Author Index Session Index M19
Martin WeiserZuse Institute [email protected]
AbstractThe biodomain model of cardioelectric excitation consists of a reaction-diffusionequation, an elliptic algebraic constraint, and a set of pointwise ODEs. Fast reactionenforces small time steps, such that for common mesh sizes the reaction-diffusionequation is easily solved implicitly due to a dominating mass matrix. In contrast,the elliptic constraint does not benefit from small time steps and requires a compa-rably expensive solution. We propose a simple delayed residual compensation thatimproves the solution of the elliptic constraint at virtually no computational costand thus alleviates the need for long iteration times.
242
GPU Accelerated Strongly Scalable Simulations of CardiacElectro-Mechanics
Session Schedule Author Index Session Index M19
Gernot PlankMedical University of [email protected]
AbstractAnatomically realistic and biophysically detailed multiscale computer models of theheart are playing an increasingly important role in advancing our understanding ofintegrated cardiac function in health and disease. Such detailed simulations, how-ever, are computationally vastly demanding, which is a limiting factor for a wideradoption of in-silico modeling. While current trends in high performance comput-ing (HPC) hardware promise to alleviate this problem, exploiting the potential ofsuch architectures remains challenging since strongly scalable algorithms are neces-sitated. Alternatively, acceleration technologies such as graphics processing units(GPUs) are being considered. While the potential of GPUs has been demonstratedin various applications, benefits in the context of bidomain simulations where largesparse linear systems have to be solved in parallel with advanced numerical tech-niques, are less clear. In previous studies, using up to 16k cores we demonstratedstrong scalability for solving the monodomain [1] as well as bidomain [2] equations.More recently, we implemented a proper domain decomposition algebraic multigridbidomain solver which can be compiled for execution on both CPUs and GPUsin distributed memory environments [3]. In this study first results are presentedon extending our solver framework to apply these methods to electro-mechanicallycoupled multi-physics simulations. Scalability results of weakly coupled electro-mechanical simulations will be presented and challenges that need to address withregard to GPU implementations will be discussed.
[1] S. Niederer, L. Mitchell, N. Smith, and G. Plank, “Simulating human cardiacelectrophysiology on clinical time-scales.” Front Physiol, vol. 2, p. 14, 2011.
[2] L. Mitchell, M. Bishop, E. Hotzl, A. Neic, M. Liebmann, G. Haase, and G. Plank,“Modeling cardiac electrophysiology at the organ level in the peta flops computingage,” AIP Conference Proceedings, vol. 1281, no. 1, pp. 407–410, 2010. [Online].Available: http://link.aip.org/link/?APC/1281/407/1
[3] Liebmann M. Hoetzl E. Mitchell L. Vigmond E.J. Haase G. Plank G. Neic, A.Accelerating cardiac bidomain simulations using graphics processing units. IEEETrans Biomed Eng, 2012 (under review).
243
An Eulerian Finite Element Method for the Simulation ofCardiomyocyte Active Contraction
Session Schedule Author Index Session Index M19
Ricardo Ruiz BaierCMCS-MATHICSE-SBEcole Polytechnique Federale de Lausanne, CH-1015 Lausanne, [email protected]
Aymen LaadhariCMCS-MATHICSE-SBEcole Polytechnique Federale de Lausanne, CH-1015 Lausanne, [email protected]
Alfio QuarteroniCMCS-MATHICSE-SBEcole Polytechnique Federale de Lausanne, CH-1015 Lausanne, [email protected]
AbstractIn this work we are interested in the modeling of the deformations of a singlecardiomyocyte surrounded by a Newtonian fluid, and their mechano-chemical inter-actions with the kinematics of calcium concentrations and active contraction. Wechoose to formulate our coupled problem in an Eulerian setting. Following [1], wetrack the boundary of the elastic body using a level set method with the particular-ity that the level set function also delivers information about the stretching of theinterface. Here we incorporate several variants on the method and on the modelitself. Namely, we employ a modified level set approach based on the impositionof additional constraints via Lagrange multipliers (see [2]). Moreover, we consideran active strain description of the activation mechanism which is based on the as-sumption of a multiplicative decomposition of the deformation gradient [3,4], andwe use slightly different models for the calcium-driven mechanical activation.
[1] G.-H. Cottet, E. Maitre, T. Milcent, Eulerian formulation and level set modelsfor incompressible fluid-structure interaction, ESAIM Math. Model. Numer. Anal.2008; 42:471–492.
[2] A. Laadhari, C. Misbah, P. Saramito, On the equilibrium equation for a general-ized biological membrane energy by using a shape optimization approach, PhysicaD 2010; 239:1567–1572.
[3] F. Nobile, A. Quarteroni, R. Ruiz-Baier, An active strain electromechanicalmodel for cardiac tissue, Int. J. Numer. Meth. Biomed. Engrg., to appear.
[4] S. Rossi, R. Ruiz-Baier, L.F. Pavarino, A. Quarteroni, An orthotropic activestrain model for the numerical simulation of cardiac biomechanics, Int. J. Numer.Meth. Biomed. Engrg., to appear.
244
Interactive Electromechanical Model of the Heart forPatient-Specific Simulation
Session Schedule Author Index Session Index M19
Maxime SermesantInria Asclepios, Sophia Antipolis, [email protected]
Hugo TalbotInria Shacra & Asclepios, Lille, [email protected]
Stephanie MarchesseauInria Asclepios, Sophia Antipolis, [email protected]
Christian DuriezInria Shacra, Lille, [email protected]
Nicholas AyacheInria Asclepios, Sophia Antipolis, [email protected]
Stephane CotinInria Shacra, Lille, [email protected]
Herve DelingetteInria Asclepios, Sophia Antipolis, [email protected]
AbstractThe software platform SOFA1 is an open-source project for interactive multi-physicssimulation of medical interventions. The objective is to design a modular frame-work so that biophysical simulations of the human body can be run in real-timeand integrated in a simulator for clinical training. Specific choices have to be madeat the modelling and implementation level in order to tackle that real-time con-straint. Moreover methods to adapt generic models to patient data are neededto obtain realistic simulations on personalised models. I will present some of thecomputational methods used in SOFA and results on cardiac electrophysiology andmechanics, as well as for cardiac catheter-based interventions. Approaches used toobtain patient-specific models will also be detailed.
1Simulation Open Framework Architecture: http://www.sofa-framework.org
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246
Mini Symposium M20Domain Decomposition and Multiscale Meth-odsSchedule Author Index Session Index
Organizers: Petter E. Bjørstad
AbstractOver the past several years, there is increased interest in Multiscale Methods and theuse of such methods to model multiscale phenomena. At the same time, it has alsobeen discovered that some of these methods share important properties with DomainDecomposition. Having different motivation and purpose, these methods often focuson the interaction between a coarse and a fine level. This mini-symposium will inviteparticipants from both communities and focus on methods where shared propertiesperhaps may be analyzed in a common framework.
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M20 Schedule
Domain Decomposition and Multiscale Methods
Schedule Author Index Session Index
Date: Wednesday, June 27Time: 16:00-17:45Location: PetriChairman: Petter E. Bjørstad
16:05-16:30 : Talal RahmanAlternative Coarse Spaces for Additive Schwarz Methods for MultiscaleElliptic ProblemsAbstract
16:30-16:55 : Juan GalvisDomain Decomposition and Multiscale Methods for High-contrast EllipticEquationsAbstract
16:55-17:20 : Robert ScheichlWeak Approximation Properties of Elliptic Projections with FunctionalConstraintsAbstract
17:20-17:45 : Rui DuTwo-Level Additive Schwarz Methods with Adaptive Sampling CoarseSpaces for Multiscale Problems in High Contrast MediaAbstract
248
Alternative Coarse Spaces for Additive Schwarz Methods forMultiscale Elliptic Problems
Session Schedule Author Index Session Index M20
Talal RahmanFaculty of Engineering, Bergen University College Nygrdsgaten 112, 5020 BERGEN,[email protected]
AbstractIn this talk, we will discuss alternative ways to construct the coarse problem for themultiscale problem, without +the need of introducing a coarse triangulation. Suchcoarse problems will have coarse spaces which are based on the partition +of unityand the energy minimizing principle, and at the same time offer certain flexibilityin constructing robust methods +for generally highly varying coefficients.
249
Domain Decomposition and Multiscale Methods forHigh-contrast Elliptic Equations
Session Schedule Author Index Session Index M20
Juan GalvisTexas A&M [email protected]
AbstractWe explore the relations between the construction of coarse spaces in “NumericalHomogenization/MsFEM (Multiscale Finite Element Methods)” and the construc-tion of coarse solvers in multilevel domain decomposition methods. With coarsespace that use local spectral information we construct robust dd methods and ro-bust multiscale (upscaling) approximations. We present representative numericalexamples.
250
Weak Approximation Properties of Elliptic Projections withFunctional Constraints
Session Schedule Author Index Session Index M20
Robert ScheichlUniversity of Bath, United [email protected]
Panayot S. VassilevskiLawrence Livermore National Laboratory, [email protected]
Ludmil ZikatanovPenn State University, [email protected]
AbstractThis paper is on the construction of energy-minimizing coarse spaces that obey cer-tain functional constraints and can thus be used, for example, to build robust coarsespaces for elliptic problems with large variations in the coefficients. In practice theyare built by patching together solutions to appropriate local saddle point or eigen-value problems. We develop an abstract framework for such constructions, akin toan abstract Bramble–Hilbert-type lemma, and then apply it in the design of coarsespaces for discretizations of PDEs with highly varying coefficients. The stabilityand approximation bounds of the constructed interpolant are in the weighted L2norm and are independent of the variations in the coefficients. Such spaces can beused, for example, in two-level overlapping Schwarz algorithms for elliptic PDEswith large coefficient jumps generally not resolved by a standard coarse grid or fornumerical upscaling purposes. Some numerical illustration is provided.
251
Two-Level Additive Schwarz Methods with Adaptive SamplingCoarse Spaces for Multiscale Problems in High Contrast Media
Session Schedule Author Index Session Index M20
Rui DuDepartment of Informatics, University of Bergen Postboks 7800, NO-5020 BERGEN,[email protected]
AbstractIn this talk we introduce novel coarse spaces, within the framework of two leveladditive Schwarz methods for elliptic problems with highly varying coefficients. Thebasis functions of coarse spaces are constructed by discrete a-harmonic extensionof the standard piecewise linear basis functions over sampling cells that cover thehigh conductivity regions. We prove that the proposed preconditioners convergeindependent of the contrast in the coefficients. Numerical results are also presentedto confirm this.
252
Contributed Talks C1Contact and Mechanics ProblemsSchedule Author Index Session Index
Date: Monday, June 25Time: 10:30-12:15Location: I51Chairman: Christian Rey
10:35-11:00 : Jaroslav HaslingerA Domain Decomposition Algorithm for Contact Problems with Coulomb’sFrictionAbstract
11:00-11:25 : Brahim NouiriMultiplicative Schwarz Method for Nonlinear Quasi-Variational Inequalitiesand their Application in Contact MechanicsAbstract
11:25-11:50 : Ihor I. ProkopyshynParallel Domain Decomposition Methods for Multibody Contact Problemsof Nonlinear ElasticityAbstract
11:50:12:15 : Alexandros MarkopoulosTotal FETI Method in Mechanics ProblemsAbstract
253
A Domain Decomposition Algorithm for ContactProblems with Coulomb’s FrictionSession Schedule Author Index Session Index C1
Jaroslav HaslingerDepartment of Numerical Mathematics, Charles University [email protected]
Radek KuceraDepartment of Mathematics and Descriptive Geometry, VSB-TU [email protected]
Taoufik SassiLaboratoire de Mathematiques Nicolas Oresme, Universite de Caen Basse- Nor-mandie, [email protected]
AbstractContact problems take an important place in the computational mechanics. Manynumerical procedures have been proposed in engineering literature. The discretiza-tion of such problems leads to very large and ill-conditioned systems. Domain de-composition methods represent a possible remedy how to overcome this difficulty.This contribution deals with an iterative method for numerical solving contact prob-lems with Coulomb friction for two elastic bodies Ω1 and Ω2. Each iterative step ofour algorithm consists of four auxiliary problems separately formulated on the in-dividual bodies. First, the non-penetration and the friction conditions are treated,after that two auxiliary Neumann problems are used to ensure continuity of contactstresses along a common part of the boundary. There are more variants how torealize this idea. For instance, one can solve the Dirichlet problem on Ω1 whilethe non-penetration and friction conditions are considered on Ω2. Alternatively,the friction and non-penetration conditions may be decomposed between bodies Ω1
and Ω2, respectively. Finally, the Gauss-Seidel splitting can be used for problemson Ω1 and Ω2 to get smaller subproblems in terms of the normal and tangentialcontact stresses.Numerical experiments indicate scalability of the variants of our algorithms for somechoices of the relaxation parameter.
254
Multiplicative Schwarz Method for NonlinearQuasi-Variational Inequalities and their Application inContact MechanicsSession Schedule Author Index Session Index C1
Brahim NouiriLaboratory of Computer Science and Mathematics,Faculty of Science,University ofLaghouat, Road Ghardaa, BP 37G,Laghouat (03000), [email protected]
Benyattou BenabderrahmaneLaboratory of Computer Science and Mathematics,Faculty of Science,University ofLaghouat, Road Ghardaa, BP 37G,Laghouat (03000), [email protected]
AbstractWe present and analyze subspace correction method for the solution of nonlinearquasi-variational inequalities and apply these theoretical results to non smooth con-tact problems in nonlinear elasticity with slip-rate dependent friction. We introducethis method in a Hilbert space, prove that it is globally convergent and give errorestimates. In the context of finite element discretization, where my method turnsout to be one-and two-level Schwarz methods, we specify their convergence rateand its dependence on the discretization parameters and conclude that our meth-ods converge optimally. Transferring this results to frictional contact problems, wethus can overcome the mesh dependence of some fixed-point schemas which are com-monly employed for contact problems with slip-rate dependent Tresca and Coulombfriction.
255
Parallel Domain Decomposition Methods forMultibody Contact Problems of Nonlinear ElasticitySession Schedule Author Index Session Index C1
Ihor I. ProkopyshynPidstryhach Institute for Applied Problems of Mechanics and Mathematics, Lviv,[email protected]
Ivan I. DyyakIvan Franko National University of Lviv, [email protected]
Rostyslav M. MartynyakPidstryhach Institute for Applied Problems of Mechanics and Mathematics, Lviv,[email protected]
Ivan A. ProkopyshynIvan Franko National University of Lviv, [email protected]
AbstractWe propose on continuous level several classes of parallel domain decompositionmethods for solution of multibody contact problems of nonlinear elasticity.We consider the contact problems of two types. The problems of the first type arethe problems of frictionless unilateral contact between several nonlinear elastic bod-ies. The problems of the second type are the problems of contact between severalnonlinear elastic bodies through intermediate nonlinear Winkler layers. We givevariational formulation of the first type problems in the form of nonlinear varia-tional inequality on the closed convex set, and of the second type problems in theform of nonlinear variational equation in Hilbert space. For the problems of thefirst type the penalty method is used to reduce the nonlinear variational inequalityon the convex set to the nonlinear variational equation in the whole space. Theexistence of a unique solution of all variational problems is shown, and the strongconvergence of the solution of penalty variational equation to the solution of initialvariational inequality is proved.For solution of nonlinear variational equations, which correspond to contact prob-lems of both types, we propose such stationary and nonstationary iterative methods,which lead to the domain decomposition. In each iterative step of these methodswe have to solve in a parallel way some linear variational equations in separatesubdomains, which correspond to linear elasticity problems with Robin boundaryconditions on possible contact areas and with some additional volume forces. Weprove theorems about the strong convergence of these domain decomposition meth-ods.
256
Total FETI Method in Mechanics ProblemsSession Schedule Author Index Session Index C1
Alexandros MarkopoulosVSB-Technical University of Ostrava, Czech [email protected]
Tomas KozubekVSB-Technical University of Ostrava, Czech [email protected]
Tomas BrzobohatyVSB-Technical University of Ostrava, Czech [email protected]
AbstractWe briefly review the TFETI based domain decomposition methodology adapted tothe solution of 2D and 3D multibody contact problems with Tresca and Coulombfriction. Recall that TFETI enforces the imposed Dirichlet boundary conditions(given displacements) by Lagrange multipliers, so that all the subdomains are float-ing and their kernels are a priori known. We present our in a sense optimal algo-rithms for the solution of resulting quadratic programming problems together withtheir powerful ingredients such as massively parallel and scalable implementationand stable generalized inverse computation. The unique feature of these algorithmsis their capability to solve the class of quadratic programming problems with spec-trum in a given positive interval in O(1) iterations. The theory yields the errorbounds that are independent of conditioning of constraints and the results are valideven for linearly dependent equality constraints. The performance is demonstratedby the solution of difficult real world problems as analysis of the yielding clampconnection of the arc steel support used in mining industry, the roller bearing ofwind generator etc.
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258
Contributed Talks C2Contact and Mechanics ProblemsSchedule Author Index Session Index
Date: Monday, June 25Time: 16:00-18:10Location: I51Chairman: Leonardo Baffico
16:05-16:30 : Daniel ChoiA Posteriori Error Estimates and Domain Decomposition Algorithm forContact ProblemsAbstract
16:30-16:55 : Vincent VisseqScalability Study of the NonSmooth Contact Domain DecompositionMethod (NSCDD)Abstract
16:55-17:20 : Geoffrey DesmeureA Mixed Domain Decomposition Method for Structural Assemblies withInterface Tractions Represented in H1/2
Abstract
17:20-17:45 : Julien RitonA Robin Domain Decomposition Algorithm for Contact Problem with givenFrictionAbstract
17:45-18:10 : Philippe KaramianA Numerical Implementation of Homogenization Technique to Evaluate theEffective Mechanical Properties of Reinforced Polymer Composites in theFrame of Domain DecompositionAbstract
259
A Posteriori Error Estimates and DomainDecomposition Algorithm for Contact ProblemsSession Schedule Author Index Session Index C2
Daniel ChoiLaboratoire de Mathematiques Nicolas Oresme, Universite de Caen Basse-Normandie,France
Laurent GallimardLaboratoire Electronique, Mecanique, Energitique, Universite Paris Ouest Nanterre-La Defence, France
Taoufik SassiLaboratoire de Mathematiques Nicolas Oresme, Universite de Caen Basse-Normandie,France
AbstractWe consider an iterative domain decomposition method associated to a finite el-ement method to approximate a unilateral contact problem between two elasticbodies. Each iterative step consists of a Dirichlet problem for the one body, acontact problem for the other one and two Neumann problems to coordinate con-tact stresses. We present a global error estimator that takes into acount as well ofthe error introduced by finite element analysis as the error commited by the itera-tive resolution of the domain decomposition algorithm. The control of these errorssources is a key point in order to introduce adaptive techniques based on errorsindicators that estimate the contribution of each source of error.Some numerical results are presented, showing the pratical efficiency of the estima-tor.
260
Scalability Study of the NonSmooth Contact DomainDecomposition Method (NSCDD)Session Schedule Author Index Session Index C2
Vincent VisseqLMGC - Universite Montpellier [email protected]
Alexandre MartinLMGC - Universite Montpellier [email protected]
Pierre AlartLMGC - Universite Montpellier [email protected]
David DureisseixLaMCoS - INSA [email protected]
AbstractFrom few years a FETI-like Domain decomposition called NonSmooth Contact Do-main Decomposition (NSCDD) is developed to handle simulations of large discretesystems, such as a railway ballast submitted to cyclic loading or the behavior ofthe Nımes arena and Arles aqueduct subjected to seismic loading. A SequentialMultidomain Implementation of the NSCDD method is performed to analyze its op-timal efficiency. Moreover, parallel implementation in 2D and 3D, on the LMGC90platform, using MPI library is herein studied as regards of two communicationsscheme. Those studies highlight influences of specificities of discrete systems as:corner grains, large rigid bodies and reorganizations of the contact network.
261
A Mixed Domain Decomposition Method forStructural Assemblies with Interface TractionsRepresented in H1/2
Session Schedule Author Index Session Index C2
Geoffrey DesmeureLMT Cachan, 61 av. du President Wilson, 94230 Cachan [email protected]
Pierre GosseletLMT Cachan, 61 av. du President Wilson, 94230 Cachan [email protected]
Christian ReyLMT Cachan, 61 av. du President Wilson, 94230 Cachan [email protected]
Philippe CrestaEADS Innovation Works France Computational Structural Mechanics,18 rue Marius Terce, 31025 Toulouse [email protected]
AbstractMechanical industries’ increasing need of liability in numerical simulations leads toevermore fine and complex models. They must take into account complicated physi-cal behaviours among which non-linarities and multiple physical scales are recurrent.With the aim of modelling large complex structural assemblies, we propose a non-overlapping mixed domain decomposition method based on a LaTIn-type iterativesolver. It relies on splitting the studied domain into substructures and interfaces,both being able to bear mechanical behaviours so that interface behaviours likeperfect cohesion, contact or delamination can be modelled. The associated Uzawasolver enables to treat at small scales nonlinear phenomena. The method, thus,can be easily parallelized and scalabilty is then ensured by a coarse problem. Onemain difficulty of the method is that its efficiency strongly depends on parameters,like search directions, varying with the geometry and materials of the structure. Itseffectiveness also relies on the choice of appropriate mechanical fields representationand discretization , among which interface tractions. The method presented usesthe Riesz representation theorem to represent interface tractions in H1/2 so thatit is possible to discretize them accordingly to the displacements. This allows toachieve an independance of convergence from mesh. The presentation will also evi-dence that high precision can be reached in few iterations. It will assess the methodfor perfect and contact interfaces that are the basic behaviours found in structuralassemblies.
262
A Robin Domain Decomposition Algorithm forContact Problem with given FrictionSession Schedule Author Index Session Index C2
Julien RitonLaboratoire de Mathematique Nicolas Oresme, Universite de Caen Basse Nor-mandie, [email protected]
Taoufik SassiLaboratoire de Mathematique Nicolas Oresme, Universite de Caen Basse Nor-mandie, [email protected]
AbstractDevelopment of numerical methods for the solution of contact problems is a chal-lenging task whose difficulty lies in the non-linear conditions for non-penetrationand friction. In this contribution, we propose and study a Robin domain decompo-sition algorithm to approximate contact problems with Tresca friction beween twobodies. Indeed this algorithm combines, in the contact zone, the Dirichlet and Neu-mann boundaries conditions (Robin boundary condition). The advantage consistsin solving in parallel the same variational inequality in each body. By numericalexperiments, we illustrate that the algorithm is mesh independent for a suitablechoice of parameters.
263
A Numerical Implementation of HomogenizationTechnique to Evaluate the Effective MechanicalProperties of Reinforced Polymer Composites in theFrame of Domain DecompositionSession Schedule Author Index Session Index C2
Philippe KaramianUniversite de Caen Basse-Normandie, UMR 6139 LMNO, F-14032 Caen, [email protected]
Willy LeclercUniversite de Caen Basse-Normandie, UMR 6139 LMNO, F-14032 Caen, [email protected]
AbstractIn the frame of the domain decomposition we present an approach to evaluatethe effective mechanical properties of short fibres reinforced polymer composites in2 or 3 dimension. In the framework of double-scale homogenization method onemust build a representative volume element (RVE) which must be large enoughto represent the mechanical aspect of the medium. We have implemented a fullyautomated fast and reliable model to conceive RVEs which are big enough especiallyin the three-dimensional case. Despite the computer performances, these RVEscannot be directly used to evaluate the Young and shear moduli in the frame workof the homogenization process. Due to the large number of degree of freedom thecalculation of the elasticity tensor and thus the compliance tensor is out of reach.The aim of this talk is to present some numerical results obtained with the help ofdirect Schur complement method and parallel programming. Actually, in the 2D-case the RVE is split in 4 partitions for instance and each portion is individuallytreated by given cpu. The main difficulties stay first in the mesh generation andsecond in the treatment of periodic boundary conditions for which we present anatural way to treat them even though the method has its benefit and inconveniencebut affordable.
264
Contributed Talks C3Optimized Schwarz MethodsSchedule Author Index Session Index
Date: Wednesday, June 27Time: 10:30-12:15Location: I50Chairman: Kevin Santugini
10:35-11:00 : Florence HubertOptimized Schwarz Algorithms for Finite Volume SchemesAbstract
11:00-11:25 : Lahcen LaayouniOn the Algebraic Optimized Schwarz methods (AOSM): Performances andApplicationsAbstract
11:25-11:50 : Erell JamelotDomain Decomposition for the Neutron SPN EquationsAbstract
11:50:12:15 : Frederic MagoulesA Stochastic-based Optimized Schwarz Method for the GravimetryEquations on GPU ClustersAbstract
265
Optimized Schwarz Algorithms for Finite VolumeSchemesSession Schedule Author Index Session Index C3
Florence HubertLATP-Aix-Marseille [email protected]
Laurence HalpernLAGA-University Paris [email protected]
AbstractWe present a finite volume discretization of Ventcell type optimized Schwarz algo-rithms for advection-diffusion equations. The approach includes a wide range ofconvection approximation. We prove convergence of the discrete algorithm withenergy estimates. Numerical illustrations of the properties of the scheme and of theSchwarz algorithm will be given.
266
On the Algebraic Optimized Schwarz methods(AOSM): Performances and ApplicationsSession Schedule Author Index Session Index C3
Lahcen LaayouniSchool of Science and Engineering, Al Akhawayn University, Avenue Hassan II,53000. P.O. Box 1630, Ifrane, [email protected]
Daniel SzyldDepartment of Mathematics, Temple University (038-16) 1805 N. Broad Street,Philadelphia, Pennsylvania 19122-6094, [email protected]
AbstractRecently, the Algebraic Optimized Schwarz Methods (AOSM) have been introducedto solve differential equations. The idea of AOSM has been inspired from thewell-known optimized Schwarz methods (OMS). The AOSM methods are based onthe modification of the block matrices associated to the transmission conditionsbetween sub-domains. The transmission blocks are replaced by modified blocksto improve the convergence of the corresponding methods. In the optimal case,the convergence can be achieved in two iterations. We are interested in how thealgebraic optimized Schwarz methods, used as preconditioner solvers, perform insolving partial differential equations. We are also interested in their asymptoticbehavior with respect to change in problems parameters. We will present differentnumerical simulations corresponding to different type of problems in two- and three-dimensions
267
Domain Decomposition for the Neutron SPNEquationsSession Schedule Author Index Session Index C3
Erell JamelotCommissariat a l’Energie Atomique et aux Energies Alternatives, CEA Saclay,DEN/DANS/DM2S/SERMA/LLPR, 91191 Gif-sur-Yvette Cedex, [email protected]
Patrick Ciarlet Jr.POEMS Laboratory, CNRS-INRIA-ENSTA UMR 7231, ENSTA ParisTech 32, Boule-vard Victor, 75739 Paris Cedex 15, [email protected]
Anne-Marie BaudronCommissariat a l’Energie Atomique et aux Energies Alternatives, CEA Saclay,DEN/DANS/DM2S/SERMA/LLPR, 91191 Gif-sur-Yvette Cedex, [email protected]
Jean-Jacques LautardCommissariat a l’Energie Atomique et aux Energies Alternatives, CEA Saclay,DEN/DANS/DM2S/SERMA/LLPR, 91191 Gif-sur-Yvette Cedex, [email protected]
AbstractStudying numerically the steady state of a nuclear core reactor is expensive, in termsof memory storage and computational time. In order to address both requirements,one can use a domain decomposition method, implemented on a parallel computer.We present here such a method for the neutron SPN equations, which are an approx-imation of the transport neutron equation. This method is based on the Schwarziterative algorithm with Robin interface conditions to handle communications (seeP.-L. Lions (1988) in: Glowinski, R., et al. Eds.). From a computational point ofview, this method is rather easy to implement. We will analyse the domain decom-position from the continuous equations to their discretization, and we will detail onhow to optimize its convergence (see F. Nataf, F. Nier, (1997) in Numer. Math.75). Finally, we will give some numerical results in a realistic 3D configuration.Computations are carried out with the MINOS solver (see A.-M. Baudron, J.-J.Lautard (2007) in Nuclear Science and Engineering 155), which is a multigroupSPN solver of the APOLLO3r2 neutronics code. Numerical experiments show thatthe method is robust and efficient, and that our choice of the Robin parameters issatisfactory.
2APOLLO3 is a trademark registered in France
268
A Stochastic-based Optimized Schwarz Method forthe Gravimetry Equations on GPU ClustersSession Schedule Author Index Session Index C3
Frederic MagoulesApplied Mathematics and Systems Laboratory, Ecole Centrale Paris, [email protected]
Abal-Kassim Cheik AhamedApplied Mathematics and Systems Laboratory, Ecole Centrale Paris, [email protected]
AbstractLow order, sequential or non-massively parallel finite elements are generaly usedfor three-dimensional gravity modelling as well as popular block semi-analyticalsolutions. Here, in order to obtain better gravity anomaly solutions in heteroge-neous media, we solve the gravimetry problem using massively parallel high orderfinite elements method on hybrid multi-CPU/GPU clusters. Parallel algorithmswhich are well suited to such architectures have to be designed, such as optimizedSchwarz methods. In this paper, theoretical and numerical results are presentedto define a new stochastic-based optimization procedure for the Schwarz methodfor the gravimetry equations. Then, we use graphical cards processors units to ac-celerate the solution. In hybrid architectures (CPU-GPU), each subdomain couldeasily be allocated to one single processor, each processor dealing with the itera-tion of the optimized Schwarz method, whereas at each iteration the solution ofthe equations on each subdomain is performed on the GPU card. Unfortunately,to obtain high speed-up, several implementation optimizations should be carrefullyperformed, such as data transfert between CPU and GPU, matrix data storage,etc. In this paper, we investigate, describe and present the optimizations we havedeveloped for finite elements problems, leading to better efficiency than existinglibrairies (CUSP, CUSPARSE, etc) for the solution on each subdomain. Numer-ical experiments performed on a reallistic case of Chicxulub crater, demonstratesthe robustness, efficiency and high speed-up of the proposed method and of itsimplementation on massive multi-CPU/GPU architecture.
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Contributed Talks C4Domain Decomposition for Helmholtz Equa-tionSchedule Author Index Session Index
Date: Tuesday, June 26Time: 14:45-15:35Location: MarkovChairman: Ana Alonso Rodriguez
14:45-15:10 : Chris StolkDomain Decomposition for Helmholtz Equations with PML BoundaryConditionsAbstract
15:10-15:35 : Dalibor LukasBEM–based Domain Decomposition MethodsAbstract
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Domain Decomposition for Helmholtz Equations withPML Boundary ConditionsSession Schedule Author Index Session Index C4
Chris StolkUniversity of [email protected]
AbstractWe study a domain decomposition method for finite difference discretizations ofthe variable coefficient Helmholtz equation. On the subdomains we use a generalpurpose multifrontal sparse LU decomposition. In 2-D, the complexity of the multi-frontal method is O(N3/2) operations and O(N logN) storage space, where N is thenumber of unknowns. This is favorable, but can still lead to memory limitations forvery large problems on a serial machine. In 3-D the multifrontal method requiresO(N2) operations and O(N4/3) storage space, which can be very costly for largeproblems. By domain decomposition these issues can be addressed. We derive anew type of boundary conditions at the boundaries between the subdomains basedon a careful analysis of the constant coefficient problem. The resulting method isused as a preconditioner in the GMRES iterative solution method. We present nu-merical experiments for a number of different choices for the variable coefficient inthe Helmholtz equation. In these experiments the preconditioned iterative methodconverged in only a few iterations for both small and large problem sizes.
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BEM–based Domain Decomposition MethodsSession Schedule Author Index Session Index C4
Dalibor LukasVSB–Technical University of [email protected]
Petr KovarVSB–Technical University of [email protected]
Lukas MalyVSB–Technical University of [email protected]
AbstractWe present two scalable domain decomposition methods that rely on boundary in-tegral formulations.First, we consider Galerkin boundary element method (BEM) accelerated by meansof hierarchical matrices (H-matrices) and adaptive cross approximation (ACA). Thisleads to almost linear complexity O(n log n) of a serial code, where n denotes thenumber of boundary nodes or elements. Once the setup of an H-matrix is done,parallel assembling is straightforward via a load-balanced distribution of admis-sible (far-field) and nonadmissible (near-field) parts of the matrix to N concur-rent processes. This traditional approach scales the computational complexity asO((n log n)/N). However, the boundary mesh is shared by all processes. We pro-pose a novel method, which leads to memory scalability O((n log n)/
√N), which is
optimal due to the dense matter of BEM matrices. The method relies on our recentresults in cyclic decompositions of directed graphs. Each process is assigned to asubgraph and to related boundary submesh. We apply the method to 3d Helmholtzproblem. The parallel scalability is documented on a distributed memory computerup to 91 cores.In the second part of the talk, we propose a boundary element preconditioner to theprimal domain decomposition method introduced by Bramble, Pasciak, and Schatzin 1986 under the name DD1. Their Schur complement system on the skeleton actsas Dirichlet-to-Neumann operator discretized by finite elements. We decompose theskeleton into faces, build the related preconditioner as a block diagonal (face-wise)BEM approximation of Neumann-to-Dirichlet operator and accelerate the diagonalblocks by H-matrices and ACA. The method is robust with respect to coefficientjumps.
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Contributed Talks C5Heterogeneous Problems and Coupling Meth-odsSchedule Author Index Session Index
Date: Tuesday, June 26Time: 14:45-15:35Location: TuringChairman: Eric Blayo
14:45-15:10 : Jonathan YouettA Time Discretization for a Heterogeneous Knee Model involving ContactProblemsAbstract
15:10-15:35 : Manel TayachiDesign of a Schwarz Coupling Method for a Dimensionally HeterogeneousProblemAbstract
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A Time Discretization for a Heterogeneous KneeModel involving Contact ProblemsSession Schedule Author Index Session Index C5
Jonathan YouettFreie Universitat Berlin, Department of Mathematics and Computer [email protected]
Oliver SanderFreie Universitat Berlin, Department of Mathematics and Computer [email protected]
Ralf KornhuberFreie Universitat Berlin, Department of Mathematics and Computer [email protected]
AbstractWe present a dynamical heterogeneous knee model consisting of geometrically exactelastic continua and Cosserat rods. The non-penetration condition between femurand tibia leads to a dynamical large-deformation contact problem. Due to thedifference of the mechanical systems at hand we apply different time integrationschemes for the rods and the continua. The continua are discretized using a contact-stabilized midpoint rule and we use the Energy–Momentum method for the Cosseratrods. The resulting coupled spatial problems are then solved using a Dirichlet–Neumann algorithm. A monotone multigrid SQP method is used as the solver forthe local nonlinear contact subproblems. We investigate the energy behaviour ofthe time discretization and illustrate our results by numerical experiments.
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Design of a Schwarz Coupling Method for aDimensionally Heterogeneous ProblemSession Schedule Author Index Session Index C5
Manel TayachiLaboratoire Jean Kuntzmann, INRIA, [email protected]
Eric BlayoLaboratoire Jean Kuntzmann, Universite Joseph Fourier, [email protected]
Antoine RousseauLaboratoire Jean Kuntzmann, INRIA, [email protected]
AbstractWhen dealing with simulation of complex physical phenomena, one may have to cou-ple several models which levels of complexity and computational cost are adaptedto the local behavior of the system. In order to avoid heavy numerical simulations,one can use the most complex model only at locations where the physics makes itnecessary, and the simplest ones - usually obtained after simplifications - everywhereelse. Such simplifications in the models may involve a change in the geometry andthe dimension of the physical domain. In that case, one deals with dimensionallyheterogeneous coupling.Our final objective is to derive an efficient coupling strategy between 1-D/2-D shal-low water equations and 2-D/3-D Navier-Stokes system. As a first step in thisdirection and in order to identify the main questions that we will have to face, wewill present in this talk a preliminary study in which we couple a 2-D Laplace equa-tion with non symmetric boundary conditions with a corresponding 1-D Laplaceequation. We will first show how to obtain the 1-D model from the 2-D one byintegration along one direction, by analogy with the link between shallow waterequations and the Navier-Stokes system. Then, we will focus on the design ofan efficient Schwarz-like iterative coupling method. We will discuss the choice ofboundary conditions at coupling interfaces. We will prove the convergence of suchalgorithms and give some theoretical results related to the choice of the location ofthe coupling interface, and the control of the error between a global 2-D referencesolution and the 2-D coupled one. These theoretical results will be illustrated nu-merically. Finally we will present some first numerical results of a test case couplinga 3-D Navier-Stokes system with a 1-D shallow water model.This work is performed in the context of a collaboration with EDF R&D.
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Contributed Talks C6Heterogeneous Problems and Coupling Meth-odsSchedule Author Index Session Index
Date: Tuesday, June 26Time: 10:30-11:50Location: I51Chairman: Rolf Krause
10:35-11:00 : Marco DiscacciatiDomain-Decomposition Preconditioners for the Darcy-Stokes ProblemAbstract
11:00-11:25 : Marina VidrascuMatched Asymptotic Expansion and Domain Decomposition for an ElasticStructureAbstract
11:25-11:50 : Christian EngwerHeterogeneous Coupling for Implicitly Described DomainsAbstract
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Domain-Decomposition Preconditioners for theDarcy-Stokes ProblemSession Schedule Author Index Session Index C6
Marco DiscacciatiLaboratori de Calcul Numeric (LaCaN),Escola Tecnica Superior d’Enginyers de Camins, Canals i Ports de Barcelona (ET-SECCPB),Universitat Politecnica de Catalunya (UPC BarcelonaTech),Campus Nord UPC - C2, E-08034 Barcelona, [email protected]
AbstractIn this talk we present some preconditioning strategies based on domain decomposi-tion theory for the finite element approximation of the coupled Darcy-Stokes prob-lem. In particular, we consider possible interface equations associated to the Darcy-Stokes problem that can be obtained using either the classical Steklov-Poincare ap-proach or a new one that we call “augmented”.We compare the different linear systems arising from those methodologies and wecharacterize suitable preconditioners of additive or multiplicative type that require,at each iteration, to solve independently the fluid and/or the porous-media sub-problems.We provide estimates for the condition number of the preconditioned systems andwe discuss the effectiveness of the different approaches taking also into account theircomputational cost.Finally, we illustrate the behavior of the proposed methods on several numericaltests of physical relevance.
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Matched Asymptotic Expansion and DomainDecomposition for an Elastic StructureSession Schedule Author Index Session Index C6
Marina VidrascuINRIA EPI [email protected]
Giuseppe GeymonatLMS, UMR-CNRS 7649 Ecole [email protected]
Sofiane HendiliI3M, UMR-CNRS 5149 U Montpellier and [email protected]
Francoise KrasuckiI3M, UMR-CNRS 5149 U [email protected]
AbstractThe study of the behavior of a structure made of a material with a large numberof heterogeneities using a standard finite element method is very expensive becausethe characteristic size of a heterogeneity is much smaller than the whole structure.An alternative approach is to use a multi-scale method such as those defined in theframework of matched asymptotic expansions. We assume that the displacementand stress fields admit two asymptotic expansions, one far from the heterogeneities(the outer one) another one close (the inner one). Basically the matched asymptoticexpansions allow to replace the initial problem by a set of new ones where the layer ofheterogeneities is replaced by a surface (in 3d) or a line in (2d) on which particularjumping conditions for displacement and stresses are defined. We show that theorder 0 outer problem is independent of the heterogeneities. For the first orderouter problem the transmission coefficients are given by several elementary innerproblems posed on a representative cell. The number of such problems dependson the nature of the heterogeneities. They all have the same structure, and canbe solved by domain decomposition of Neumann-Neumann or Robin-Robin type.More precisely on on each subdomain, we will look for a solution ui which takes thefollowing form ui = wi+βiz
i with βi two real numbers conveniently chosen. In thisdecomposition zi is the solution of a standard problem and takes into account thegap in displacements while wi is continuous on the interface but there is a gap inthe stresses. When writing the interface problem for wi using the Steklov-Poincareoperator we notice that it differs from a standard elasticity problem only in theright hand side. Numerical results will validate this approach by comparing thesolution obtained using this method to the standard solution of the problem.
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Heterogeneous Coupling for Implicitly DescribedDomainsSession Schedule Author Index Session Index C6
Christian EngwerInstitute for Computational und Applied Mathematics, University of [email protected]
Sebastian WesterheideInstitute for Computational und Applied Mathematics, University of [email protected]
AbstractModern imaging techniques yield high quality information of complex shaped mi-croscopic structures. The unfitted discontinuous Galerkin method (UDG) offers anapproach to solve PDEs on implicitly described sub-domains, e.g. obtained usingmicro-CT imaging, without the need to construct a geometry-resolving mesh. Thedomain description uses a level-set based formulation; still sub-domain boundariesare incorporated explicitly. While UDG allows an easy application to multi-domainproblems, these techniques are not sufficient for many biological application, whichinvolve coupling of volume and surface processes.We discuss an extension of the UDG method to incorporate processes on mani-fold in a heterogeneous domain-decomposition framework. UDG constructs basisfunctions from a simple background mesh and restrict the support according to theactual sub-domain boundaries, i.e. the implicitly prescribed domain. Using theexplicit reconstruction of the implicit sub-domain boundary it is possible to couplelevel-set based surface problems on the interface with sub-domain problems. Firstresults show a model problem, coupling surface diffusion on a membrane with thesurrounding volume.
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Contributed Talks C7Domain Decomposition with Preconditionners
Schedule Author Index Session Index
Date: Tuesday, June 26Time: 16:00-17:45Location: I51Chairman: Damien Tromeur-Dervout
16:05-16:30 : Daniel SzyldAdditive Schwarz with variable weights is betterAbstract
16:30-16:55 : Feng-Nan HwangParallel Multilevel Polynomial Jacobi-Davidson Eigensolver for DissipativeAcoustic ProblemsAbstract
16:55-17:20 : Santiago BadiaOn the Scalability of Balanced Domain Decomposition Preconditioners forLarge Scale Computing: Galerkin-based and Efficient Coarse CorrectionsAbstract
17:20-17:45 : Laurent BerenguerLow-Rank Update of the Restricted Additive Schwarz Preconditioner forNonlinear SystemsAbstract
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Additive Schwarz with variable weights is betterSession Schedule Author Index Session Index C7
Daniel SzyldUnited States Temple [email protected]
Chen GreifUniversity of British [email protected]
Rees TyroneUniversity of British [email protected] Canada
AbstractWe develop a proposal for the simultaneous use of multiple preconditioners for GM-RES (or other minimal residual methods). At each step, we apply all availablepreconditioners, and choose a minimal solution from the thus constructed largersubspaces. We charachterize these subspaces, and considere “truncated” versions,where specific smaller subspaces are chosen. We apply these new ideas to the casethat each preconditioner corresponds to a local solve in an overlapping domain de-composition setting, considering additive and restricted additive Schwarz versions.Numerical results show the advantage of the proposed approach.
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Parallel Multilevel Polynomial Jacobi-DavidsonEigensolver for Dissipative Acoustic ProblemsSession Schedule Author Index Session Index C7
Feng-Nan HwangNational Central [email protected]
Yu-Fen ChengNational Central [email protected]
Tsung-Ming HuangNational Taiwan Normal [email protected]
Weichung WangNational Taiwan [email protected]
AbstractMany scientific and engineering applications require accurate, fast, robust, andscalable numerical solution of large sparse algebraic polynomial eigenvalue prob-lems (PEVPs) arising from some appropriate disretization of partial differentialequations. The polynomial Jacobi-Davidson (PJD) algorithm has been numericallyshown as a promising approach for the PEVPs to effectively find their interior spec-trum and has gained its popularity. The PJD algorithm is a subspace method,which extracts the candidate approximate eigenpair from a search space and thespace undated by embedding the solution of the correction equation at the JD it-eration. In this talk, we propose the multilevel PJD algorithm for PEVPs withemphasis on the application of the dissipative acoustic cubic eigenvalue problem.The proposed multilevel PJD algorithm is based on the Schwarz framework. Theinitial basis for the search space is constructed on the current level by using thesolution of the same eigenvalue problem, but defined on the previous coarser grid.On the other hand, a parallel efficient multilevel Schwarz preconditioner is designedfor the correction equation to enhance the scalability of the PJD algorithm, whichplays a crucial property in parallel computing for large-scale problem solved by us-ing a large number of processors. Some numerical examples obtained on a parallelcluster of computers are given to demonstrate the robustness and scalability of ourPJD algorithm.
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On the Scalability of Balanced DomainDecomposition Preconditioners for Large ScaleComputing: Galerkin-based and Efficient CoarseCorrectionsSession Schedule Author Index Session Index C7
Santiago BadiaCentre Internacional de Metodes Numerics en Enginyeria (CIMNE),Universitat Politecnica de Catalunya (UPC)[email protected]
Alberto MartınCentre Internacional de Metodes Numerics en Enginyeria (CIMNE)[email protected]
Javier PrincipeCentre Internacional de Metodes Numerics en Enginyeria (CIMNE),Universitat Politecnica de Catalunya (UPC)[email protected]
AbstractThe coarse space correction in domain decomposition sub-structuring algorithms isbasic for attaining algorithmic and weak scalability. However, this coarse correc-tion has also a negative impact, since its assembly/solution requires global densecollectives and increases the parallelization overheads due to idling or wasted com-putation. Among these type of preconditioners, we have the BNN and BDDC al-gorithms. The BDDC preconditioner is considered superior to the BNN one since:1) it allows to solve the coarse problem only approximately and additive precondi-tioners; 2) the local solvers do not require to deal with a singular matrix and sparsedirect solvers can be used; 3) the coarse problem is sparser than the one of BNNand its sparsity pattern is similar to the one of the original matrix.We rehabilitate BNN with respect to point 1) and 2) using a similar (or even sim-pler) approach to the one of BDDC for the imposition of the local coarse constraints,allowing only approximately balanced residuals as well as the use of efficient androbust sparse direct (local) solvers. Further, the BNN algorithm has some benefitswhen compared to BDDC preconditioning. The size of the BNN coarse problem indimension 3 can be smaller than the BDDC one (using both corners and edges) andpoint 3) is not important when a sparse direct solver is used for the coarse correc-tion problem. Further, the coarse correction in BNN is of Galerkin type, allowing tosolve the coarse problem only once per iteration, even for its multiplicative version,with an impact on its scalability.We analyze the effect of the coarse solver on the weak/strong scalabity of thesepreconditioners, as well as the effect of a multiplicative Galerkin-type correction (as
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for BNN).
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Low-Rank Update of the Restricted AdditiveSchwarz Preconditioner for Nonlinear SystemsSession Schedule Author Index Session Index C7
Laurent BerenguerUniversite de Lyon, U. Lyon 1, CNRS, UMR5208, Institut Camille [email protected]
Damien Tromeur-DervoutUniversite de Lyon, U. Lyon 1, CNRS, UMR5208, Institut Camille [email protected]
AbstractThe Backward differentiation formula (BDF) discretization of nonlinear PDEs re-quires efficient Newton methods to solve the nonlinear system F (x) = 0 at each timestep. In particular Inexact Newton methods with restricted additive Schwarz (RAS)preconditioner [2] to approximate the inverse Jacobian has already been developed.Such technique involves the solution of local linear systems at each application ofthe preconditioner which can be avoided if the preconditioning matrix is computedand updated from one iteration to another. The generalized Broyden’s update ofthe RAS preconditioner is investigated here. Quasi-Newton methods update the ap-proximation the Jacobian Jk ≈ J(xk) at the inner iteration k minimizing the change||Jk − Jk−1||F under the secant condition Jk (xk − xk−1) = F (xk)− F (xk−1). Thisupdate can also be applied to J−1
k via the Sherman-Morison formula. Indeed, thereis no more linear systems to solve at each quasi-Newton iterations, but the majordrawback is the computation of J(x0)−1. A natural extension is to perform Newtonsteps, updating the preconditioner as an approximation of J−1 [1], taking advantageof the Newton methods convergence properties, and of the cheap low-rank updatingprocedure. We first propose to investigate this methodology in the domain decom-position context: updating the restricted additive Schwarz preconditioner from oneNewton iteration to another. Then, a quasi-Newton algorithm is proposed, updat-ing the preconditioned system at once. Numerical experiments will be provided ondifferent formulations of the driven cavity problem.
[1] L. Bergamaschi, S. Bru., and A. Martinez. Low-rank update of preconditionersfor the inexact Newton method with spd jacobian. Mathematical and ComputerModelling, 54:1863–1873, 2011.
[2] X.-C. Cai and M. Sarkis. A restricted additive Schwarz preconditioner for generalsparse linear systems. SIAM J. Sci. Comput., 21(2):792–797 (electronic), 1999.
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Contributed Talks C8Application to Flow ProblemsSchedule Author Index Session Index
Date: Wednesday, June 27Time: 10:30-12:15Location: I51Chairman: Taoufik Sassi
10:35-11:00 : Aivars ZemitisOn Domain Decomposition Based Software Tool for Flow Simulation inContainment Pools of Nuclear ReactorsAbstract
11:00-11:25 : Leonardo BafficoA Fluid Structure Interaction Problem with Friction–Type Slip BoundaryConditionAbstract
11:25-11:50 : Francois PacullKrylov Acceleration of Schur Complement Type Iterations for LinearizedCFD Systems: a Numerical ExaminationAbstract
11:50:12:15 : Daniel LoghinInterface Preconditioners for Flow ProblemsAbstract
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On Domain Decomposition Based Software Tool forFlow Simulation in Containment Pools of NuclearReactorsSession Schedule Author Index Session Index C8
Aivars ZemitisFraunhofer ITWM, Fraunhofer Platz 1, 67663 Kaiserslautern, [email protected]
Oleg IlievFraunhofer ITWM, Fraunhofer Platz 1, 67663 Kaiserslautern, [email protected]
AbstractA large number of scenario has to be simulated in connection with nuclear reactorssafety. Here we deal with simulation of unsteady non-isothermal fluid flow for aclass of containment pools of nuclear reactors, coupled with unsteady heat transferin containment’s walls and in various obstacle. A specialized software tool is de-veloped, CoPool, which allows for a fast simulation, and at the same time providethe required accuracy in computing the thermal stratification in the fluid. Advan-tage is taken from the fact that in the general case a containment geometry canbe created using primitive geometrical objects (spheres, cylinders, parallelepipeds)and Boolean operations on them. Overlapping Domain Decomposition method,namely, additive Schwarz method, is used to solve for the heat transfer in the walls.Local orthogonal coordinate systems are attached to each primitive, and a localorthogonal grid is generated in each primitive. Finite volume method is used todiscretize the heat conduction equation in each primitive, resulting in necessity tosolve SPD problems for each primitive. The usage of local grids for each object re-sult in non-matching composite grid. Different approaches are used to approximatethe temperature on the boundary of the primitives: e.g., interpolation, moving leastsquares, etc. Further on, non-overlapping Domain Decomposition is used to decou-ple the heat transfer in the fluid part and in the walls. Special attention is paid toenergy conservation in this case. Results from tests and simulations for particularcontainments are presented and discussed.
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A Fluid Structure Interaction Problem withFriction-Type Slip Boundary ConditionSession Schedule Author Index Session Index C8
Leonardo BafficoLaboratoire de Mathematiques Nicolas Oresme, Universite de Caen - CNRS [email protected]
Taoufik SassiLaboratoire de Mathematiques Nicolas Oresme, Universite de Caen - CNRS [email protected]
AbstractWe study a simple model of the stationary interaction between a rigid tube and aviscous fluid. In the original model, introduced in the late ’80’s, the fluid motionis governed by the Stokes equation and the displacements of the tube’s section aredetermined using a simple algebraic equation. In the present work, instead of theclassic no-slip condition for the fluid velocity on the fluid-solid interface, we supposethat the fluid can slip on this interface if the shear stress reach a threshold value. Inthe first part of this talk we will present the fluid-solid interaction problem and wewill show an existence result based on a fixed point argument. In the second part,we will expose the numerical approximation (based on Lagrange multipliers) imple-mented using FreeFEM++, and we will point out some drawbacks of this mixedapproach that could be corrected using a domain decomposition approach. Finally,we will address possible extensions of this model (e.g., to consider a deformablestructure, unsteady case), both from theoretical (existence/uniqueness result) andnumerical (domain decomposition/stabilization technics) point of view.
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Krylov Acceleration of Schur Complement TypeIterations for Linearized CFD Systems: a NumericalExaminationSession Schedule Author Index Session Index C8
Francois [email protected]
Stephane [email protected]
AbstractIn the field of CFD optimization, efficient, robust and accurate algebraic solversfor large, sparse and real linear systems arising from the full linearization of thecompressible Navier-Stokes equations, including the turbulence model, are critical,whether in the discrete direct or adjoint approaches. To this regard, parallel hy-brid direct/iterative solvers with local direct solvers (e.g. multifrontal sparse LU),algebraic domain decomposition preconditioners (e.g. RAS) and iterative Krylovsubspace solvers (e.g. GMRES) are often used. However, these last two types ofmethods are penalized by the fact that the linear systems considered are severelylacking most of the enjoyable properties such as symmetry, diagonal dominance,definite positiveness and well-conditioning, especially in the convection-dominatedcases. The associated convergence rate is often being strongly diminished by thetight anisotropic coupling between neighboring sub-domains as well as the largenumber of artificial interfaces that need to be crossed by the information during theprocess, when no coarse approximation of the global solution is employed. Becausethe linkage between sub-domains is crucial, we investigate in the present paper theKrylov acceleration of iterations on Schur complement type systems, that is, ofsequences of unknowns located at each sub-domain external interface nodes. Al-though this approach might not be competitive, the interface system exhibit somedistinct spectral properties from the global one, which we will indirectly deducefrom the GMRES convergence behavior. It is also smaller, despite the fact that theinterface/interior nodes ratio is rather large in 3D with high-order discretizations.Note that the Schur complement type matrix does not have to be built explicitly:its product with a given vector can be deduced from a single Schwarz iterate withhomogeneous local problems, provided that the initial guess is carefully chosen andthat some constant vector terms are added. This means that the same numericaltools as for the traditional hybrid linear solvers can be used for the investigation.Finally, the indirect effect of the global partitioning on the interface system will bestudied.
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Interface Preconditioners for Flow ProblemsSession Schedule Author Index Session Index C8
Daniel LoghinUniversity of [email protected]
AbstractWe describe a class of constrained interface preconditioners for the Stokes andthe Oseen equations. Our approach involves constructing preconditioners basedon discrete interpolation norms via a suitable Krylov approximation method. Weshow that the coercivity and continuity of the bilinear forms induced by the interfaceoperator provide mesh-independent preconditioners for both the Stokes and Oseenproblems. Numerical results using standard test problems are included to illustratethe procedure and verify the optimality of the proposed solver technique.
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Contributed Talks C9Multidomains and Time Domain Decomposi-tionSchedule Author Index Session Index
Date: Wednesday, June 27Time: 16:00-17:45Location: I50Chairman: Xiao-Chuan Cai
16:05-16:30 : Chao YangParallel Implicit Method for Phase-Field ProblemsAbstract
16:30-16:55 : Felix KwokAnalysis of a Predictor-Corrector Method with Many SubdomainsAbstract
16:55-17:20 : Martin CermakTotal-FETI Domain Decomposition Method for Solving Elasto-PlasticProblem with ContactAbstract
17:20-17:45 : Damien Tromeur-DervoutNon Linear Boundary Conditions for Time Domain Decomposition MethodAbstract
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Parallel Implicit Method for Phase-Field ProblemsSession Schedule Author Index Session Index C9
Chao YangInstitute of Software, Chinese Academy of [email protected]
Xiao-Chuan CaiDepartment of Computer Science, University of Colorado, Boulder, [email protected]
David KeyesKAUST and Columbia [email protected]
Michael PerniceIdaho National Laboratory, [email protected]
AbstractPhase-field modeling has found numerous applications in materials science. Dueto the multiscale nature, partial differential equations arising in many phase-fieldmodels are typically high-order nonlinear parabolic PDEs containing both diffu-sive and anti-diffusive terms and are often stiff and highly ill-conditioned. In thiswork, stabilized implicit schemes with an adaptive time-stepping strategy for sometypical phase-field problems are investigated. We apply a Newton-Krylov-Schwarzalgorithm to solve the nonlinear system of equations arising at each time step.Low-order homogeneous boundary conditions for the overlapping subdomains areimposed in the Schwarz preconditioner to achieve promising convergence results.Numerical tests on a supercomputer with thousands of processor cores are providedto show the scalability of the parallel solver.
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Analysis of a Predictor-Corrector Method with ManySubdomainsSession Schedule Author Index Session Index C9
Felix KwokUniversity of [email protected]
AbstractWe analyze a predictor-corrector analogue of the Crank-Nicolson method for solv-ing linear reaction-diffusion equations in parallel on many subdomains, which hasbeen proposed by Rempe and Chopp in 2006. Unlike the standard time-step-and-precondition strategy or the waveform relaxation approach, it is not necessary toiterate to convergence; one advances in time simply by predicting the interface val-ues using forward Euler, solving for the remaining variables parallel, and finallycorrecting the interface values with a backward Euler step. We show that themethod is unconditionally stable and formally second order in time for a fixed spa-tial grid. However, if we refine the spatial and temporal grid simultaneously, themethod is only second order when ∆t = O(hα) for α ≥ 3/2, unlike the classicalCrank-Nicolson method, which requires α ≥ 1 only. We also show that the error isinversely proportional to the diameter of the subdomains.
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Total-FETI Domain Decomposition Method forSolving Elasto-Plastic Problem with ContactSession Schedule Author Index Session Index C9
Martin CermakVSB-Technical University of [email protected]
Stanislav SysalaInstitute of Geonics AS CR, [email protected]
Alexandros MarkopoulosVSB-Technical University of [email protected]
AbstractIn this paper, we consider a rate-independent elasto-plastic problem with isotropichardening and with the Signorini contact boundary conditions. We assume the vonMises plastic criterion and the associated plastic flow rule. The time discretiza-tion of the problem is based on the implicit Euler method. The correspondingone-time-step problem is formulated in incremental form with respect to unknowndisplacement. We approximate the problem by the finite element method. Forsuch a problem, we propose an “external” algorithm based on a linearization ofthe elasto-plastic stress-strain relation by the corresponding tangential operator. Itmeans, that in each iteration, we solve a variational inequality of the first kind, thatis similar to an elastic problem with contact. The considered algorithm has a localquadratic convergence and coincides with the semismooth Newton method if thecontact boundary conditions are not considered. The arising variational inequal-ities can be formulated as problems of quadratic programming with equality andinequality constraints. Such problems are solved by the “internal” algorithm, whichis based on the SMALSE method in combination with the Total-FETI domain de-composition method [Dostal, Z., Kozubek, T., Markopoulos, A., Brzobohaty, T.,Vondrak, V., Horyl, P.: Theoretically supported scalable TFETI algorithm for thesolution of multibody 3D contact problems with friction. Computer Methods in Ap-plied Mechanics and Engineering 205-208 (1) , pp. 110-120, 2012]. The main ideaof the algorithm is based on enforcing the contact boundary conditions, the Dirich-let boundary conditions and the intersubdomain continuity conditions by Lagrangemultipliers. The algorithm enables a parallel implementation and has parallel scal-ability. The elasto-plastic problem with contact was implemented into the MatSollibrary in parallel Matlab environment [Kozubek, T., Markopoulos, A., Brzobohaty,T., Kucera, R., Vondrak, V. and Dostal, Z.: MatSol - MATLAB efficient solvers forproblems in engineering, http://matsol.vsb.cz/]. We illustrate the performance ofour algorithm on benchmarks for 2D and 3D.
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Non Linear Boundary Conditions for Time DomainDecomposition MethodSession Schedule Author Index Session Index C9
Damien Tromeur-DervoutUniversite de Lyon, U. Lyon1, CNRS, UMR5208 Institut Camille [email protected]
Patrice LinelDepartment of Biostatistics and Computational Biology, University of RochesterPatrice [email protected]
AbstractWe developed parallel time domain decomposition methods to solve systems ofODEs based on the Aitken-Schwarz (C. R. Math. Acad. Sci. Paris, 349(15-16):911-914, 2011) or primal Schur complement domain decomposition methods (Advancesin Parallel Computing 19:75-82,2010). The method claims the transformation ofthe initial value problem in time defined on [0, T ] into a boundary values problemin time. To overcome the lack of knowledge of the boundary value at t = T , thetime interval is symmetrized with respect to T in order to integrate backward theODEs with a symmetric scheme and to come back to t = 0 where the value ofthe time derivative is known. Then classical domain decomposition methods canbe applied with no overlapping time slices. A Dirichlet (respectively Neumann)boundary condition at the begin (respectively end) of the time slice are imposedfor the local problem. In this talk, we will present some recent developments of themethod where we show that Dirichlet-Neumann boundary conditions for non linearODEs are not optimal. Then we will propose non linear boundary conditions thataccelerate the convergence and simplify the methodology also.
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Contributed Talks C10Application to Flow ProblemsSchedule Author Index Session Index
Date: Wednesday, June 27Time: 16:00-17:45Location: I51Chairman: Frederic Magoules
16:05-16:30 : Jyri LeskinenDistributed Shape Optimization Using the Coupling of DDM of NonlinearFlows and GDM of Shapes on a Hybrid CPU/GPU PlatformAbstract
16:30-16:55 : Mohamed Khaled GdouraDomain Decomposition for Stokes Problem with Tresca Friction: AugmentedLagrangian ApproachAbstract
16:55-17:20 : Thu Huyen DaoA Schur Complement Method for Two-Phase Flow ModelsAbstract
17:20-17:45 : Guillaume HouzeauxSchur or not Schur: not so sureAbstract
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Distributed Shape Optimization Using the Couplingof DDM of Nonlinear Flows and GDM of Shapes on aHybrid CPU/GPU PlatformSession Schedule Author Index Session Index C10
Jyri LeskinenUniversity of Jyvaskyla, Jyvaskyla, [email protected]
Jacques PeriauxUniversity of Jyvaskyla, Jyvaskyla, FinlandCIMNE, Barcelona, [email protected]
AbstractIn this work we present the latest results of a new distributed one-shot method usingDDM/GDM techniques with Nash algorithms. In inverse shape design problems,algorithmic convergence can be improved significantly by splitting the design vectorinto subvectors and computing efficiently the Nash equilibrium solution of the asso-ciated multi-objective problem using competitive (Nash) games due to the reductionof the search space. With DDM techniques reconstruction of nonlinear flows canalso be formulated as a Nash game where a traditional decomposition method isaugmented with an optimizer which minimizes the deviation of the solutions at theoverlapped interfaces of subdomains. Considerable speed-ups in wall-clock time canbe achieved by combining these two games into one global “one shot” Nash gamewhere the flow is reconstructed simultaneously with the geometry.
The improved efficiency and design quality are presented and discussed from thecomputational results of the new method on aeronautical configurations like multielement airfoils subjected to large geometric changes using a hybrid CPU/GPU plat-form consisting of several CPU nodes and state-of-the-art GPUs and its extension tomore complex geometries and nonlinear mathematical flow modeling emphasized.
302
Domain Decomposition for Stokes Problem withTresca Friction: Augmented Lagrangian ApproachSession Schedule Author Index Session Index C10
Mohamed Khaled GdouraSL Rasch, Visualization Department Kesslerweg 22, Leinfelden-Oberaichen, [email protected]
Mohamed Ali IpopaUniversite des Sciences et Techniques de Masuku Ecole Polytechnique de MasukuBP, Franceville, [email protected]
Jonas KokoLIMOS, Universite Blaise-Pascal, CNRS UMR 6158, [email protected]
Taoufik SassiLaboratoire de Mathematiques Nicolas Oresme, Universite de Caen Basse-Normandie,[email protected]
AbstractDevelopment of numerical methods for the solution of Stokes system with slipboundary conditions (Tresca friction conditions) is a challenging task whose dif-ficulty lies in the non-linear conditions. Such boundary conditions have to be takeninto account in many situations arising in practice.We propose and study a domain decomposition method for the Stokes problem withslip boundary conditions which treats the constraint of velocity at the interfaces byLagrangian multiplier method. Based on augmented Lagrangian techniques, it firstrewrites the original global minimization problem as a saddle-point problem andthen solve it by an Uzawa block relaxation method involving three supplementaryconditions . The numerical realization of such problems will be discussed and resultsof a model example will be shown.
303
A Schur Complement Method for Two-Phase FlowModelsSession Schedule Author Index Session Index C10
Thu Huyen DaoCEA–Saclay, DEN, DM2S, STMF, LMEC, F–91191 Gif–sur–Yvette, FranceAppl. Mat. and Syst. Lab., Ecole Centrale Paris, 92295 Chatenay–Malabry, [email protected]
Michael NdjingaCEA–Saclay, DEN, DM2S, STMF, LMEC, F–91191 Gif–sur–Yvette, [email protected]
Frederic MagoulesAppl. Mat. and Syst. Lab., Ecole Centrale Paris, 92295 Chatenay–Malabry, [email protected]
AbstractIn this talk, we will report our recent efforts to apply a Schur complement methodfor nonlinear hyperbolic problems. We use the finite volume method and an implicitversion of the Roe approximate Riemann solver. With the interface variable intro-duced in [1] in the context of single phase flows, we are able to simulate two-fluidmodels ([2]) with various schemes such as upwind, centered or Rusanov. Moreover,we introduce a scaling strategy to improve the condition number of both the inter-face system and the local systems. Numerical results for the isentropic two-fluidmodel and the compresible Navier-Stokes equations in various 2D and 3D configura-tions and various schemes show that our method is robust and efficient. The scalingstrategy considerably reduces the number of GMRES iterations in both interfacesystem and local system resolutions. Comparisons of performances with classicaldistributed computing with up to 512 processors are also reported.
[1] T.H. Dao, M. Ndjinga, F. Magoules: A Schur Complement Method for theCompressible Navier-Stokes Equations Accepted in Proceeding of 20th InternationalConference on Decomposition Methods.
[2] M. Ndjinga, A. Kumbaro, F. De Vuyst, P. Laurent-Gengoux: Numerical sim-ulation of hyperbolic two-phase flow models using a Roe-type solver Nucl. Eng.Design, 238(2008), 2075-2083.
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Schur or not Schur: not so sureSession Schedule Author Index Session Index C10
Guillaume HouzeauxBarcelona Supercomputing Center - Centro Nacional de [email protected]
Frederic MagoulesApplied Mathematics and Systems Laboratory, Ecole Centrale Paris, [email protected]
Beatriz EguzkitzaBarcelona Supercomputing Center - Centro Nacional de [email protected]
AbstractWe present in this work a comparison between various algebraic solvers for solvingfluid flow problems in bio-mechanics, namely the airflow in the human large airwaysand the brain hemodynamics, which geometries come from medical imaging. Wefocus on the pressure equation coming from a stabilized finite element method forthe Navier-Stokes equations, which leads to a symmetric positive definite matrix.The use of complex geometries, like the ones envisaged in this work share, a com-mon feature. They involve very elongated domains (airways, hundereds of brainarteries), which render the solution of such problems tricky. In fact, low frequencymodes are barely damped by classical solvers and render the use of multigrid ordeflation based solver uncircumventable for such cases. In this talk, we compareclassical parallelized iterative solvers, Schur-based parallel solver and algebraic op-timized Schwarz-based solver for the efficient solution of this problem. The solversconsidered are tested on a supercomputer, using up to 1024 CPU’S, for large meshesinvolving more than 100 million of elements. The comparison is carried out usingthe following criteria: CPU time, speedup, efficiency and convergence rate.
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Contributed Talks C11Time Parallel - Parareal MethodsSchedule Author Index Session Index
Date: Thursday, June 28Time: 10:30-11:50Location: I50Chairman: Michael Minion
10:35-11:00 : Noha Makhoul-KaramRatio-Based Parallel Time IntegrationAbstract
11:00-11:25 : Daniel RuprechtHybrid Space-Time Parallel Solution of Burger’s EquationAbstract
11:25-11:50 : Rolf KrauseA Massively Space-Time Parallel N -Body SolverAbstract
307
Ratio-Based Parallel Time IntegrationSession Schedule Author Index Session Index C11
Noha Makhoul-KaramAUB [email protected]
Nabil NassifAUB [email protected]
Jocelyne ErhelIRISA [email protected]
AbstractBecause time-integration of time-dependent problems is inherently sequential, timeparallelism aims mainly at reducing the computational time of some real-time evo-lutionary problems and may be done through predictor-corrector schemes.In a previous paper, we have presented a new approach for solving some of suchproblems in a time-parallel way that uses an end-of-slice condition, together witha rescaling methodology, and automatically decomposes the time-domain. Thisapproach is possible when the end-of-slice values of the solution exhibit a ratio-property and it results in a Ratio-based Parallel Time Integration (RaPTI) al-gorithm that has been successfully tested on some particular bounded problems.It differs from other time-domain decomposition techniques by the fact that thetime-slices are not predefined, they are rather deduced gradually (together withthe solution) as a measure of the progress, within a local time, of the prediction-correction procedure.We now apply the rescaling method onto initial value problems having an explosivesolution, in infinite time. We show how a relevant choice of the end-of-slice condi-tion and the time-rescaling factor might lead to rescaled systems having an “asymp-totic similarity” (i.e. uniform convergence to a limit problem). When exploited byRaPTI algorithm, such asymptotic similarity provides much better predictions andenhances the relevance of RaPTI that consists mainly of (i) the little sequentialcomputations it involves (predictions and corrections are done in parallel), (ii) therelatively low communication cost it induces and (iii) the similarity of the compu-tation on all slices yielding similar computational times on all processors. Hence,significant speed-ups are achieved. This is illustrated on two problems: a non-lineardiffusion-reaction problem having an explosive solution, and a membrane problemhaving an oscillatory and explosive solution.
Keywords: Initial value problems, rescaling, end-of-slice condition, uniform convergence,
parallel time-integration.
308
Hybrid Space-Time Parallel Solution of Burger’sEquationSession Schedule Author Index Session Index C11
Daniel RuprechtInstitute of Computational Science, University of Lugano, [email protected]
Roberto CroceInstitute of Computational Science, University of Lugano, [email protected]
Rolf KrauseInstitute of Computational Science, University of Lugano, [email protected]
AbstractWe present a hybrid shared-memory/distributed-memory implementation of Parareal[1] combined with spatial domain decomposition and investigate its performance forthe two-dimensional Burger’s equation using up to 128 cores. Time-parallel meth-ods, like Parareal, provide an additional direction for parallelization beyond spatialparallelization by decomposing the computational domain [2]. Space-time parallelschemes combining Parareal or other time-parallel methods with spatial paralleliza-tion can potentially help to utilize the anticipated massive numbers of cores availablein upcoming high-performance computing systems.In our approach, Parareal is implemented using shared-memory while the spatial do-main decomposition uses MPI. This eliminates the need to communicate full volumedata in the iteration of Parareal. We demonstrate that the hybrid [3] space-timeparallel implementation can provide speedup beyond the saturation of the spatialparallelization. Details on the organization of the MPI communication of ghost-cellsfor the multi-threaded code are presented and the overhead resulting from the needto exchange data for multiple points in time simultaneously is addressed.
[1] J. Lions and Y. Maday and G. Turinici, A ”parareal” in time discretization ofPDE’s, C. R. Acad. Sci. – Ser. I – Math. 332, 661–668, 2001.
[2] Y. Maday and G. Turinici, The parareal in time iterative solver: A further di-rection to parallel implementation, LNCSE 40, 441–448, 2005.
[3] R. Rabenseifner et al., Hybrid MPI/OpenMP Parallel Programming on Clustersof Multi-core SMP Nodes, 17th Eurom. Int. Conf. on Parallel, Distributed andNetwork-based processing, 427–436, 2009.
309
A Massively Space-Time Parallel N-Body SolverSession Schedule Author Index Session Index C11
Rolf KrauseISC, Universita della Svizzera italiana, Lugano (Switzerland)[email protected]
Robert SpeckICS, Universita della Svizzera italiana, Lugano (Switzerland)[email protected]
Daniel RuprechtICS, Universita della Svizzera italiana, Lugano (Switzerland)[email protected]
Matthew EmmettDepartment of Mathematics, University of North Carolina, Chapel Hill (USA)[email protected]
Mathias WinkelJSC, Forschungszentrum Julich GmbH, Julich (Germany)[email protected]
Paul GibbonJSC, Forschungszentrum Julich GmbH, Julich (Germany)[email protected]
Michael MinionDepartment of Mathematics, University of North Carolina, Chapel Hill (USA)[email protected]
AbstractWe present a novel space-time parallel version of the hybrid Barnes-Hut tree codePEPC that uses the Parallel Full Approximation Scheme in Space and Time (PFASST)algorithm to perform parallelized time integration. For small problem sizes strongscaling of the spatial parallelization within PEPC naturally becomes saturated. Tofurther reduce time-to-solution, an additional direction of parallelism is needed.For this purpose, PFASST uses an iterative ODE method based on deferred cor-rection sweeps within a parareal-like iterative time-parallel strategy. These sweepsare combined with multiple space-time discretization levels to reduce the compu-tational cost per iteration, hence relaxing the strict bound on parallel effciency ofparareal. We illustrate the potential of space-time parallel algorithms to extend theintrinsic strong scaling limit of purely spatial parallelism, which is an indispensablerequirement to efficiently use the fast-growing number of cores in upcoming HPCarchitectures.
310
Contributed Talks C12Optimization Methods/Probabilistic Methods
Schedule Author Index Session Index
Date: Friday, June 29Time: 10:30-12:15Location: I51Chairman: Hui Zhang
10:35-11:00 : Andreas LangerDomain Decomposition Methods for a Class of Non-smooth ConvexVariational ProblemsAbstract
11:00-11:25 : Firmim Andzembe OkoubiDomain Decomposition with Nesterov’s Minimization MethodAbstract
11:25-11:50 : Francisco BernalA Meshfree Scheme for PDEs on Large Domains using Probabilistic DomainDecompositionAbstract
11:50:12:15 : Samia RiazA Domain Decomposition Method for Elliptic Variational InequalitiesAbstract
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Domain Decomposition Methods for a Class ofNon-smooth Convex Variational ProblemsSession Schedule Author Index Session Index C12
Andreas LangerInstitute for Mathematics and Scientific Computing,University of Graz,Heinrichstraße 36,A-8010 Graz, [email protected]
Michael HintermullerDepartment of Mathematics,Humboldt-University of Berlin,Unter den Linden 6,10099 Berlin, [email protected]
AbstractIn the last decades, in the literature, there have been introduced many different ap-proaches and algorithms for minimizing non-smooth convex variational problems.These standard techniques are iterative-sequentially formulated and therefore notable to solve large scale simulations in acceptable computational time. For suchlarge problems we need to address methods that allow us to reduce the problem toa finite sequence of subproblems of a more manageable size, perhaps computed byone of the standard techniques. With this aim, we introduce domain decomposi-tion methods for non-smooth convex variational problems. A prominent example ofsuch problems is the minimization of the total variation. The main idea of domaindecomposition is to split the space of the initial problem into several smaller sub-spaces. By restricting the function to be minimized to the subspaces, a sequenceof local problems, which may be solved easier and faster than the original prob-lem, is constituted. Then the solution of the initial problem is obtained via thesolutions of the local subproblems by gluing them together. In the case of domaindecomposition for non-smooth and non-additive problems the crucial difficulty isthe correct treatment of the interfaces of the domain decomposition patches. Dueto the non-smoothness and non-additivity, one encounters additional difficulties inshowing convergence of more general subspace correction strategies to global mini-mizers. Nevertheless, we are able to propose an implementation of overlapping andnon-overlapping domain decomposition algorithms for non-smooth, non-additive,and convex functionals with the guarantee of convergence to a minimizer of theoriginal functional and the monotonic decay of the energy. We provide severalnumerical experiments, showing the successful application of the algorithms.
312
Domain Decomposition with Nesterov’s MinimizationMethodSession Schedule Author Index Session Index C12
Firmim Andzembe OkoubiUniversite Clermont-Ferrand [email protected]
Jonas KokoUniversite Clermont-Ferrand [email protected]
AbstractA nonoverlapping domain decomposition with N subdomains can be formulated asthe following constrained minimization problem
min
N∑i=1
Ji(ui) (5)
[uij ] = 0, (6)
where
• Ji are convex functions,
• [uij ] = (ui − uj)|Γij, the solution gap across the interface Γij = ∂Ωi ∩ ∂Ωj .
Nesterov’s gradient descent method [Nesterov Y., A method of solving a convex pro-gramming problem with convergence rate of O(1/k2), Soviet. Math. Dokl. 27 (2),372-376 (1983)] is a convex optimization method with convergence rate of O(1/k2),where k is the current iteration. We apply this method to solve problem (5)-(6).Since the Nesterov method is a first order method, we do not solve linear systemsduring the iterative process. The saving of computational time can therefore besignificant.
313
A Meshfree Scheme for PDEs on Large Domainsusing Probabilistic Domain DecompositionSession Schedule Author Index Session Index C12
Francisco BernalInstituto Superior Tecnico, [email protected]
Juan A. AcebrnInstituto Superior Tecnico, Lisbonjuan.acebron@@ist.utl.pt
AbstractProbabilistic Domain Decomposition (PDD) is a novel approach to the numericalsolution of partial differential equations (PDEs) on a parallel computer. Like manyother state-of-the-art domain decomposition methods, PDD splits the domain alonginternal interfaces into non-overlapping subdomains, which are handed over to aPDE solver. The hallmark of PDD is a previous step in which the solution tothe PDE is interpolated along the artificial interfaces, exploiting the stochasticrepresentation of the PDE through a Monte Carlo approach. Therefore, each of thesubdomains holds a well-posed problem that can be solved independently, and hencewithout inter-processor communication. We show several examples on irregulargeometries which use collocation meshfree methods based on radial basis functions(RBFs) as the PDE subdomain solver.
314
A Domain Decomposition Method for EllipticVariational InequalitiesSession Schedule Author Index Session Index C12
Samia RiazUniversity of [email protected]
Daniel LoghinUniversity of [email protected]
AbstractVariational inequalities have found many applications in applied science. A partiallist includes: obstacles problems, fluid flow in porous media, management science,traffic network, and financial equilibrium problems. However, solving variationalinequalities remain a challenging task as they are often subject to some set of com-plex constraints.Domain decomposition methods provide great flexibility to handle these type ofproblems. In our presentation we consider the following general variational inequal-ity Lu ≥ f in Ω,
u ≥ ψ in Ω,u = 0 on ∂Ω,
where L is an elliptic operator. We will present a non-overlapping domain decom-position formulation for variational inequalities. In our formulation, the originalproblem is reformulated into two subproblems such that the first problem is a vari-ational inequality in some subdomains Ωii and the other is a variational equality inthe complementary subdomains Ωei . This new formulation will reduce the computa-tional cost as the variational inequality is solved on a smaller region. However oneof the main challenges here is to obtain the global solution of the problem, whichis to be coupled together through an interface problem. We validate our approachon two dimensional obstacle test problems using quadratic programming.
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Contributed Talks C13FETI MethodsSchedule Author Index Session Index
Date: Friday, June 29Time: 14:00-15:45Location: MarkovChairman: Francois-Xavier Roux
14:05-14:30 : Leszek MarcinkowskiA Parallel Preconditioner for FETI-DP Method for a Crouzeix-RaviartDiscretization of an Elliptic ProblemAbstract
14:30-14:55 : Ange ToulougoussouSchur Complement Methods for the Solution of the Discrete Stokes Systemwith Continuous PressureAbstract
14:55-15:20 : Hui ZhangOptimized Interface Preconditioners for the FETI MethodsAbstract
15:20-15:45 : Christian ReyStopping Criterion for FETI Solver based on an Evaluation of theDiscretization ErrorAbstract
317
A Parallel Preconditioner for FETI-DP Method for aCrouzeix-Raviart Discretization of an Elliptic ProblemSession Schedule Author Index Session Index C13
Leszek MarcinkowskiFaculty of Mathematics, University of Warsaw, Warsaw, [email protected]
Talal RahmanDepartment of Computer Engineering, Bergen University College, Bergen, [email protected]
AbstractIn the talk we discuss a parallel preconditioner for a FETI-DP problem arising froma nonconforming Crouzeix-Raviart discretization of a model second order problemwith discontinuous coefficients. Locally in the subdomains, we introduce local trian-gulations which form one global triangulation. Then the nonconforming Crouzeix-Raviart finite element space is introduced.The local interior variables are eliminatedin subdomains and next a Feti-dp problem is introduced. Finally we construct andanalyze a parallel preconditioner for the Feti-dp problem. We show that the precon-ditioner is quasi-optimal, i.e. the condition number of the preconditioned problemgrows polylogarithmically with respect to the sizes of the local meshes and is inde-pendent of jumps of coefficients in the subdomains.
318
Schur Complement Methods for the Solution of theDiscrete Stokes System with Continuous PressureSession Schedule Author Index Session Index C13
Ange ToulougoussouUniversite Pierre et Marie [email protected]
Francois-Xavier RouxOnera and Universite Pierre et Marie [email protected]
AbstractWe consider the solution of the linear system arising from the discretization of theStokes system by finite element methods with continuous pressure spaces. For asystem of large size, it is convenient to use iterative methods that take advantage ofparallel high performance computing and save memory space. FETI and BDD be-long to this class of methods and have proved efficiency for a large type of problems.They were successfully extended to the Stokes system discretized with discontinuouspressure spaces. When the approximate velocity and pressure are both continuous,the interface systems of FETI and BDD become mixed problems and have a badcondition number. We introduce a new method, the hybrid Schur method, thatcombines FETI and BDD in such a way that the interface unknowns are physi-cally homogeneous. We give some theoretical analysis of the condition number ofthe preconditioned hybrid Schur method applied to the discrete Stokes system andsome numerical comparisons with variants of FETI.
319
Optimized Interface Preconditioners for the FETIMethodsSession Schedule Author Index Session Index C13
Hui ZhangUniversity of [email protected]
Martin J. GanderUniversity of [email protected]
AbstractIt is well-known that using the Dirichlet preconditioner in the classical FETI methodleads to very nice condition number estimates. However, the Dirichlet precondi-tioner can be quite expensive, and it was proposed in the literature to use as acheaper alternative the lumped preconditioner instead. In this talk, we will shownew interface preconditioners that are optimized for rapid convergence, while theircosts are comparable to the cost of the lumped preconditioner.
320
Stopping Criterion for FETI Solver based on anEvaluation of the Discretization ErrorSession Schedule Author Index Session Index C13
Christian ReyLMT-Cachan, ENS Cachan/CNRS/UPMC/PRES UniverSud,61 av. du president Wilson, 94235 Cachan cedex, [email protected]
Pierre GosseletLMT-Cachan, ENS Cachan/CNRS/UPMC/PRES UniverSud,61 av. du president Wilson, 94235 Cachan cedex, [email protected]
Valentine ReyLMT-Cachan, ENS Cachan/CNRS/UPMC/PRES UniverSud,61 av. du president Wilson, 94235 Cachan cedex, [email protected]
Augustin Parret-FreaudLMT-Cachan, ENS Cachan/CNRS/UPMC/PRES UniverSud,61 av. du president Wilson, 94235 Cachan cedex, [email protected]
AbstractFor the last decades, three trends have grown and reinforced each other: the fastgrowth of hardware computational capacities, the requirement of finer and largerfinite element models for industrial simulations and the development of efficient com-putational strategies amongst which non-overlapping domain decomposition (DD)methods are very popular since they have proved to be scalable in many applica-tions. One main fallout lies on the verification of the discretized models in order towarranty the quality of numerical simulations (global or goal-oriented error estima-tors). In this talk, we present some of our recent works that aim at building errorestimators in a non-overlapping domain decomposition framework. We focus on theconstruction of fully parallel global error estimator based on interface admissibilityconditions. Connection with both primal (BDD) and dual (FETI) iterative domaindecomposition solver is outlined. It yields a guaranteed upper bound on the errorwhatever the state (converged or not) of the associated iterative solver. Eventually,we introduce first works that enable to separate algebraic error and discretizationerror. This leads to the definition of convergence criteria of DD iterative solversbased on discretization error estimator instead of purely algebraic criteria.
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Contributed Talks C14Time Dependent PDEs and ApplicationsSchedule Author Index Session Index
Date: Thursday, June 28Time: 10:30-12:15Location: I51Chairman: Daniel Loghin
10:35-11:00 : Petros AristidouA Schur Complement Method for DAE Systems in Power System SimulationAbstract
11:00-11:25 : Rodrigue KammogneDomain Decomposition Methods for Reaction-Diffusion SystemsAbstract
11:25-11:50 : Frederic PlumierCombining Full Transients and Phasor Approximation Models in PowerSystem Time SimulationAbstract
11:50-12:15 : David CherelDomain Decomposition For Stokes Equations Using Waveform RelaxationMethodAbstract
323
A Schur Complement Method for DAE Systems inPower System SimulationSession Schedule Author Index Session Index C14
Petros AristidouUniversity of Liege, [email protected]
Davide FabozziUniversity of Liege, [email protected]
Thierry Van CutsemFNRS and University of Liege, [email protected]
AbstractThe power system networks in North America and Europe are the largest man-made interconnected systems in the world. Many power system applications relyon time consuming dynamic simulations of large-scale power systems in order tooptimize the operation and ensure the reliability of the electricity network. Dynamicsimulations of power systems involve the solution of a series of initial value, stiff,hybrid DAE systems over a time window. To achieve this, the time window isdiscretized and a new DAE system is formed and solved at each time step, withinitial values taken from the previous time step solution. At each new time step, theDAE system to be solved can be different because of the discrete variables involvedin the formulation (e.g. a differential equation can become algebraic and vice versa).A non-overlapping domain decomposition is proposed to speed up the solution ofthe DAE system using the Schur Complement Method. The special structure of thephysical system helps define the domain partitioning scheme and eliminates the needfor a partitioning algorithm. It allows the formulation and solution of the reducedsystem using sparse, direct solvers to obtain the interface variables. Afterwards, theparallel evaluation of the internal subdomain variables is possible and efficient loadbalancing is achieved. Numerically, the method shows no convergence degradationwhen compared to the integrated method, which is traditionally used for solvingpower system DAEs. The aspects of decomposition, solution and optimization ofthe algorithm for the specific problem are discussed and results from the applicationof the DDM on realistic power system models are presented.
324
Domain Decomposition Methods forReaction-Diffusion SystemsSession Schedule Author Index Session Index C14
Rodrigue KammogneUniversity of [email protected]
Daniel LoghinUniversity of [email protected]
AbstractReaction diffusion systems are an important application in the area of modern math-ematical modeling. They can be found in biology, engineering, medicine, pollutioneffect modeling, weather prediction, finance, to name but a few. However the nu-merical solution for reaction-diffusion problems remains a challenge as they oftenarise as a couple system of nonlinear PDE, which are solved on a complex domain.Domain Decomposition Methods provide a powerful and flexible tools for the nu-merical solution of reaction diffusions-systems. In our presentation we consider thefollowing generic system:
ut −∆u = f(u) in ΩBu = g on ∂Ω
where u = (u1, u2) and f ∈ R2 .We will present a rigorous formulation of non-overlapping domain decompositionmethods for reaction-diffusion systems together with the arising Steklov−Poincareoperator. Our approach is based on the well known fact that the Steklov−Poincareoperators arising in a non-overlapping DD−algorithm for scalar elliptic problemsare coercive and continuous with respect to Sobolev norms of index 1/2. Further-more, this key property will motivate the construction of new interface precondi-tioners for the Steklov−Poincare operator, which leads to solution techniques inde-pendent of the mesh parameters. The convergence analysis for the preconditionedGMRES together with various numerical examples are also included.
325
Combining Full Transients and Phasor ApproximationModels in Power System Time SimulationSession Schedule Author Index Session Index C14
Frederic PlumierUniversity of Liege, [email protected]
Davide FabozziUniversity of Liege, [email protected]
Christophe GeuzaineUniversity of Liege, [email protected]
Thierry Van CutsemFNRS and University of Liege, [email protected]
AbstractPower system time simulations are generally classified into two main categories,depending on the time scale of the dynamic phenomena under study. The slowesttime-scale is the so-called Phasor Approximation (PA). It focuses on the time evo-lution of the magnitude and phase angle of the various phasors. PA is simulatedusing single-phase representation of the three-phase currents and voltages in thenetwork. The other two phases are assumed to operate at the same frequency, tohave the same amplitude but mutual phase difference of ±120o. In steady-statecondition, the above assumptions are almost perfectly met, but in transient con-dition it may not be the case at all. The faster time-scale, Full Transients (FT),relies on three-phase waveform representation of the system currents and voltages,without requiring further assumptions on amplitude, frequency or phase. This leadsto a more precise simulation, although more time consuming. Generally the powersystem is simulated as a whole making use of one or the other modelling and sim-ulation methods. However, in some cases one might take advantage of a domaindecomposition into two subdomains, i.e. an FT subdomain and a PA subdomain.This is the case for instance when evaluating the effects of large disturbances in thesystem, e.g. short-circuits. Some disturbances cannot be simulated in PA programsand should then be modeled in FT, while some others can be modeled using bothmethods but would give a higher level of detail in FT. This domain decompositionmethod is applied to a simple but representative model of a small power system.Overlapping and non-overlapping approaches are thoroughly analysed.
326
Domain Decomposition For Stokes Equations UsingWaveform Relaxation MethodSession Schedule Author Index Session Index C14
David CherelLaboratoire Jean Kuntzmann, [email protected]
Eric BlayoLaboratoire Jean Kuntzmann, Universite Joseph Fourier, [email protected]
Antoine RousseauLaboratoire Jean Kuntzmann, INRIA, [email protected]
AbstractThe overall objective of this work is to construct a coupling algorithm between theNavier-Stokes equations and the Primitive Equations, which are commonly used fordescribing the ocean circulation in the open sea and are derived from the Navier-Stokes system under Boussinesq and hydrostatic approximations. Such a couplingallows for the description of the ocean dynamics from the coast to the open ocean.Two intermediate steps are first dealt with: a domain decomposition method forthe Navier-Stokes equations, and then a coupling algorithm between two Navier-Stokes equations with different aspect ratios. We shall present herein the first stepof this work, namely a domain decomposition method for the Navier-Stokes equa-tions. After a time discretization of the Navier-Stokes equations, we construct aSchwarz waveform relaxation algorithm. This algorithm uses Dirichlet-to-Neumannrelaxed operator on the interface between subdomains. Using a perfect transpar-ent operator would allow the Schwarz algorithm to converge in only two iterations.However, since such an operator is nonlocal in space, it must be approximated foractual applications. Different approximations of this operator are derived, whichgeneral forms correspond to different orders of a Taylor expansion with respect toa small parameter, and which degrees of freedom are chosen in order to optimizethe convergence rate of the algorithm.These approximations are numerically implemented in the well known test caseof the lid driven cavity. It is shown first that a projection method, widely usedfor solving the Navier-Stokes equations, is inappropriate to treat complex inter-face conditions properly. Then numerical results are given using a code solvingsimultaneously for the velocity and the pressure fields.
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Contributed Talks C15Multigrid MethodsSchedule Author Index Session Index
Date: Friday, June 29Time: 14:00-15:45Location: AmphiChairman: James Brannick
14:05-14:30 : Kab Seok KangA Parallel Multigrid Solver on a Structured Triangulation of a HexagonalDomainAbstract
14:30-14:55 : Pawan KumarParallel Aggregation Based Algebraic MultigridAbstract
14:55-15:20 : Lori BadeaMultigrid Method with Constraint Level Decomposition for VariationalInequalities with Contraction OperatorsAbstract
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A Parallel Multigrid Solver on a StructuredTriangulation of a Hexagonal DomainSession Schedule Author Index Session Index C15
Kab Seok KangHigh Level Support Team (HLST)Max-Planck-Institut fur PlasmaphysikEURATOM Association,Boltzmannstraße 2D-85748 Garching bei [email protected]
AbstractFast elliptic solvers are a key ingredient of massively parallel Particle-in-Cell (PIC)simulation codes for fusion plasmas. This applies for both, the gyrokinetic and fullykinetic models. The most efficient solver for large elliptic problems is the multigridmethod, especially the geometric multigrid method which requires detailed infor-mation of the geometry for its discretization. In our particular case, we consider aparallel geometric multigrid solver for a structured triangulation of the hexagonaldomain for an elliptic partial differential equation. Special care has been taken inoptimizing also the parallel performance by making use of the available geometricinformation. The scaling properties of the multigrid solver on massively paral-lel computers as e.g. HPC-FF and IFERC-CSC are investigated. In addition, theperformance results are compared with the results of solvers from public availablelibraries and our own implementation of the domain decomposition method.
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Parallel Aggregation Based Algebraic MultigridSession Schedule Author Index Session Index C15
Pawan KumarKU Leuven, Leuven, [email protected]
Yvan NotayUniversite Libre de Bruxelles, Bruxelles, [email protected]
AbstractDomain decomposition methods (DDM) are among the most popular methods forsolving large sparse linear systems of the form Ax = b on distributed memory archi-tectures. The method is robust when the solution at the interface are approximatedwell, or when a coarse grid solve is used, both leading to exchange of informationacross the domains which is often essential for convergence.
In this talk, we present a parallel aggregation based algebraic multigrid method.Here the graph corresponding to the matrix is partitioned using a k−way partition-ing scheme. The smooother is the Gauss-Siedel method that ignores the non-localconnections. The coarsening is done by aggregating the nodes locally based onstrength of connection and ignoring the nodes that satisfy a diagonal dominancecriterium.
The method has true black-box nature in the sense that the underlying problemmay arise from a non-PDE problem and only input required from the user is thecoefficient matrix A and the right hand side vector b.
The scalability and robustness of the method is studied on a wide range of prob-lems including convection diffusion, and problems from the Florida matrix marketcollection. The method is also compared with the existing state of the art methodson a cluster.
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Multigrid Method with Constraint LevelDecomposition for Variational Inequalities withContraction OperatorsSession Schedule Author Index Session Index C15
Lori BadeaInstitute of Mathematics of the Romanian AcademyP.O. Box 1-764, RO-014700 Bucharest, [email protected]
AbstractWe introduce a multigrid method for the variational inequalities with contractionoperators, where the closed convex set is decomposed as a sum of closed convexlevel subsets. The method is described as multigrid V -cycles, but the results holdfor other iteration types, the W -cycle iterations, for instance. We first introducethe method as a subspace correction algorithm in a general reflexive Banach space.Under an assumption on the level decomposition of the closed convex set of theproblem, we prove the convergence of the algorithm and estimate the global con-vergence rate as a functions of the number of levels. In finite element spaces, thealgorithm becomes a multigrid method for one-obstacle problems. We prove thatthe assumption we made holds, and compare the obtained convergence rate withthat of other multigrid methods in the literature. In particular, the proposed algo-rithm supplies a multigrid method for the inequalities which do not arise from theminimization of a functional.
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Contributed Talks C16Finite Element Method for Domain Decompo-sitionSchedule Author Index Session Index
Date: Friday, June 29Time: 14:00-15:45Location: PetriChairman: Marina Vidrascu
14:05-14:30 : Patrick Le TallecMultidomain Calculations with Embedded InterfacesAbstract
14:30-14:55 : Thomas DickopfEvaluating Local Approximations of the L2-Orthogonal Projection BetweenNon-nested Finite Element SpacesAbstract
14:55-15:20 : Debasish PradhanA Robin-Type Non-Overlapping Domain Decomposition Procedure forSecond Order Parabolic ProblemsAbstract
15:20-15:45 : Frederic MagoulesA Bacteria Evolution-based Partitioning Scheme for the Parallel Solution ofProblems in Computational MechanicsAbstract
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Multidomain Calculations with Embedded InterfacesSession Schedule Author Index Session Index C16
Patrick Le TallecEcole [email protected]
Gauthier FolzanEcole Polytechnique and CEA/[email protected]
Jean-Philippe PerlatCEA/[email protected]
AbstractMany simulations in fluid structure interactions or in dynamic impact problemsinvolve independent components interacting through complex interfaces. For suchproblems, standard strategies use a piecewise ALE approach with one finite elementmesh per component and adequate matching strategies. These are are difficult touse in presence of large mesh distortions or when facing topology changes. Analternative is to use a Eulerian strategy describing the different substructures on asingle grid using a single average velocity field, and developing ad hoc constitutivelaws to handle the multiphase microstructure of each element. These models areusually quite crude on the interface physics. In this context, there is a renewedinterest in models which will use a single global smooth mesh in the background, notmatching the individual components, and independent finite element velocity fieldsfor each subpart. The talk will discuss and validate such an approximation strategy.The first ingredient is a discontinuous interface reconstruction strategy based onvolumic fraction gradients. The velocity fields are defined on the background mesh,meaning that a single node possibly carries two independent velocity unknowns.A different finite element formulation is then introduced for each subdomain byrestricting the subdomain integration to the parts of the background finite elementswhich are inside the considered subdomain. The interface kinematic continuityconstraint is imposed in average inside each finite element, which introduces aninterface pressure unknown per element. In order to obtain a stable system, onemust then enrich the finite element velocities next to the interface, which is doneas in Dolbow by introducing bubble velocity shape functions on all elements withinterface constraints. A specific treatment must be developed for inertia terms, andthe convection terms are dealt with by projection of the convected quantities to thebackground mesh using exact mesh intersection.
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Evaluating Local Approximations of the L2-OrthogonalProjection Between Non-nested Finite Element SpacesSession Schedule Author Index Session Index C16
Thomas DickopfUniversity of Lugano, Institute of Computational [email protected]
Rolf KrauseUniversity of Lugano, Institute of Computational [email protected]
AbstractWe present quantitative studies of transfer operators between finite element spacesassociated with unrelated meshes. Several local approximations of the global L2-orthogonal projection are reviewed and evaluated computationally. The numericalstudies in 3D provide the first estimates of the quantitative differences betweena range of transfer operators between non-nested finite element spaces. We con-sider the standard finite element interpolation, Clement’s quasi-interpolation withdifferent local polynomial degrees, the global L2-orthogonal projection, a local L2-quasi-projection via a discrete inner product, and a pseudo-L2-projection definedby a Petrov–Galerkin variational equation with a discontinuous test space. Under-standing their qualitative and quantitative behaviors in this computational way isinteresting per se; it could also be relevant in the context of discretization and so-lution techniques which make use of different non-nested meshes. It turns out thatthe pseudo-L2-projection approximates the actual L2-orthogonal projection best.The obtained results seem to be largely independent of the underlying computa-tional domain; this is demonstrated by four examples (ball, cylinder, half torus andStanford Bunny).
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A Robin-Type Non-Overlapping DomainDecomposition Procedure for Second Order ParabolicProblemsSession Schedule Author Index Session Index C16
Debasish PradhanDept. of Applied Mathematics, Defence Institute of Advanced Technology, Girina-gar, Pune - 411 025, [email protected]
AbstractThis article deals with the analysis of an iterative non overlapping domain decom-position (DD) method for parabolic problems, using Robin-type boundary condi-tion on the inter-subdomain boundaries, which can be solved in parallel with localcommunications. The proposed iterative method allows us to relax the continuitycondition for Lagrange multipliers on the inter-subdomain boundaries. In orderto derive the corresponding discrete problem, we apply a non-conforming Galerkinmethod using the lowest order Crouzeix-Raviart elements. The convergence of theiterative scheme is obtained by proving that the spectral radius of the matrix asso-ciated with the fixed point iterations is less than 1. For ∆t = O(h2), we derive theupper bound of the rate of convergence which is of order 1 - O(h1/2H−1/2), whereh is the finite element mesh parameter, H is the maximum diameter of the subdo-mains and ∆t is the time step. The numerical experiments confirm the theoreticalresults established in this paper.
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A Bacteria Evolution-based Partitioning Scheme forthe Parallel Solution of Problems in ComputationalMechanicsSession Schedule Author Index Session Index C16
Frederic MagoulesApplied Mathematics and Systems Laboratory, Ecole Centrale Paris, [email protected]
Abilash KrishnanDepartment of Mechanical Engineering, National Institute of Technology Karnataka,Surathkal, [email protected]
AbstractFor parallel execution a finite element mesh must be divided, in other words parti-tioned, so the parts can be distribute among the different processors or computers.The primary goal of mesh partitioning is to minimize communication time whilemaintaining load balance. In an explicit method, communication is associated withthe nodes that lie on the boundaries between subdomains and are shared by morethan one processor. Communication time depends on both the message sizes, whichincrease with the number of shared nodes, and the number of messages, which in-creases with the number of adjacent subdomains. In an implicit method, such asoptimized Schwarz domain decomposition methods, additional quality criteria ofthe partitioning, strongly impact the convergence of the iterative algorithm: as-pect ratio of the partitions, geometrical shape of the interface, etc. Unfortunately,some of these criteria are not implemented in standard libraries, such as METIS,SCOTCH, JOSTLE, Chaco, etc. In this paper, we propose an original partitioningprocedure based on bacteria evolution, we have developed for domain decomposi-tion methods. Bacteria propagate or reproduce most commonly by a kind of celldivision called binary fission. In binary fission, a single cell divides and two identicalcells are formed following given specific rules of the species and of the environments.Cohabitation of different bacteria species lie to an equilibrium of the global system.Inspired from this idea, and defining new bacteria evolution laws, we design a meshpartitioning methods which leads to a mesh partitioner with desirable properties,i.e. , producing subdomains of nearly equal size, and with as few nodes sharedbetween processors as is reasonably possible, and leading to excellent convergenceproperties of the optimized Schwarz algorithm. This original algorithm is presentedtogether with several applications on computational mechanics problems which out-lines the robustness, efficiency and quality of the partition, compared with existinglibrairies.
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Contributed Talks C17Non-matching Grids/Nonconforming Discretiza-tionSchedule Author Index Session Index
Date: Friday, June 29Time: 14:00-15:45Location: TuringChairman: Guillaume Houzeaux
14:05-14:30 : Kirill Pichon GostafFinite Element Analysis of Multi-Component Assemblies: CAD - basedDomain DecompositionAbstract
14:30-14:55 : Ajit PatelMortar Finite Element Methods for Hyperbolic ProblemsAbstract
14:55-15:20 : Eliseo Chacon VeraStabilization of a FETI-DP Mortar Method for the Stokes ProblemAbstract
15:20-15:45 : Beatriz EguzkitzaA Chimera Method applied to Computational Solid DynamicsAbstract
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Finite Element Analysis of Multi-ComponentAssemblies: CAD - based Domain DecompositionSession Schedule Author Index Session Index C17
Kirill Pichon GostafLaboratoire Jacques-Louis Lions, UPMC, [email protected]
Olivier PironneauLaboratoire Jacques-Louis Lions, UPMC, [email protected]
Francois-Xavier RouxONERA, [email protected]
AbstractComputer Aided Design (CAD) and Finite Element modeling are standards in aconcept to manufacture industrial chain. Realistic simulations require not onlyhuge computational resources, but may last several days or even years. We proposeto apply the domain decomposition methodology (DDM) to carry out numericalsimulations of multi-component CAD assemblies. The novelty of our research isthe CAD - based domain decomposition. We consider design parts as independentsubdomains, and we use assembly topology to define regions, where the interfaceboundary conditions should be applied. There are two key motivations for em-ploying this strategy: an attempt to regularize the mathematical models and theirparallel computations, and to facilitate finite element management of the essentialCAD data. Obviously, our principal objective is to adapt domain decompositionmethods and to develop consistent numerical schemes, which are inherently paralleland, therefore, are perfectly suited for high-performance computing (HPC). In or-der to enforce continuity of the global domain decomposition solution, the interfaceconditions have to be settled on the boundaries between all adjacent subdomains.These conditions are imposed iteratively and may vary in time. We examine theconvergence rate of the global numerical solution, which is extremely sensitive andstrongly depends on these interface conditions, especially in case of either non-matching or geometrically discontinuous discretizations. The Dirichlet-Neumann,the Neumann-Neumann and the FETI methods, as well as their mortar based ex-tensions are studied in this work. We detail the proposed CAD - based domaindecomposition strategy with numerical experiments, and we focus attention on theessence of a parallel implementation.
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Mortar Finite Element Methods for HyperbolicProblemsSession Schedule Author Index Session Index C17
Ajit PatelThe LNM Institute of Information [email protected]
AbstractIn this article, the mortar finite element method is used for spatial discretizationand a finite difference scheme is used for time discretization of a class of hyperbolicproblems. Optimal error estimates in L2- and H1-norms for both semidiscrete andfully discrete schemes are discussed. The results of numerical experiments supportthe theoretical results obtained.
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Stabilization of a FETI-DP Mortar Method for theStokes ProblemSession Schedule Author Index Session Index C17
Eliseo Chacon VeraDpto. Ecuaciones Diferenciales y Analisis Numerico, Facultad de Matematicas,Universidad de Sevilla, Tarfia sn. 41012 [email protected]
Tomas Chacon RebolloDpto. Ecuaciones Diferenciales y Analisis Numerico, Facultad de Matematicas,Universidad de Sevilla, Tarfia sn. 41012 [email protected]
AbstractWe couple a non standard FETI-DP Mortar with stabilization to the discretizationof the Stokes equations. The flux term across interfaces is computed via the Rieszrepresentation and this gives freedom to choose matching or not triangulations atinterfaces. Moreover, the computational cost is reduced when stabilization tech-niques are also used. Theoretical analysis as well as some numerical tests will bepresented. This research extends the ideas recently introduced in [1] and [2] forelliptic problems and in [3] for Stokes equations.
[1] C. Bernardi, T. Chacon Rebollo and E. Chacon Vera, A FETI method with amesh independent condition number for the iteration matrix Computer Methods inApplied Mechanics and Engineering, Vol 197/13-16 pp 1410–1429., 2008.
[2] E. Chacon Vera, A continuous framework for FETI-DP with a mesh independentcondition number for the dual problem Computer Methods in Applied Mechanicsand Engineering, Vol 198, pp 2470–2483., 2009.
[3] E. Chacon Vera, D. Franco Coronil and A. Martınez Gavara A non standardFETI-DP mortar method for Stokes Problem Submitted.
Acknowledgments: Research partially funded by Spanish government MEC Re-search Project MTM2009-07719.
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A Chimera Method applied to Computational SolidDynamicsSession Schedule Author Index Session Index C17
Beatriz EguzkitzaBarcelona Supercomputing Center, [email protected]
Guillaume HouzeauxBarcelona Supercomputing Center, [email protected]
Denny TjahjantoMadrid Institute for Advanced Studies, [email protected]
Mariano VazquezBarcelona Supercomputing Center, [email protected]
Antoine JerusalemMadrid Institute for Advanced Studies, [email protected]
AbstractWe present in this work a Chimera method applied to Computational Solid Dynam-ics (CSD). The Chimera method was originally design as a technique for simplifyingthe mesh generation, by glueing overlapping and non-conforming meshes together.During many years, it has been almost excusively applied to CFD, although theconcept is quite general. We propose here to extend the method to CSD, wherevery few examples are given in the literature. In CFD, the mehtod is useful to treatmoving objects, each object having an independent attached mesh. In the contextof CSD, the method can fairly simplify the meshing and enables to add, remove ormove components to the geometry without having to remesh the global geometry. Itcan also be used for optimization purpose where the free degrees of freedom are thepositions of the objects. The first step of the method is the classical hole cutting.The hole cutting consists in removing elements from the background mesh locatedinside the patch meshes. The second step consists in imposing some conditionson the subdomain interfaces in order to obtain continuous solution and flux acrossthem. The proposed method impose Dirichlet conditions in an implicit way so theresulting algorithm does not introduce any additional iterative loop and is fullyimplicit. The method is also inherently parallel. We present some validation andapplication cases demonstrating the efficiency of the method on a supercomputer.
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Contributed Talks C18FETI MethodsSchedule Author Index Session Index
Date: Tuesday, June 26Time: 11:50:12:15Location: I51Chairman: Rolf Krause
11:50:12:15 : K. C. ParkA Simple Explicit-Implicit FETI Transient Analysis AlgorithmAbstract
Date: Thursday, June 28Time: 11:50-12:15Location: I50Chairman: Michael Minion
11:50-12:15 : Ulrich LangerFETI-Solvers for Non-standard Finite Element Equations based onBoundary Integral OperatorsAbstract
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A Simple Explicit-Implicit FETI Transient AnalysisAlgorithmSession Schedule Author Index Session Index C18
K. C. ParkDepartment of Aerospace Engineering Sciences, University of Colorado atBoulder, CO 80309-429, USA., and Division of Ocean Systems Engineering,KAIST, Daejeon 305-701, Republic of [email protected] A. GonzalezEscuela Superior de Ingenieros, Camino de los Descubrimientos s/n, E-41092Seville, [email protected]
AbstractA simple explicit-implicit FETI (AFETI-EI) algorithm is presented for partitionedtransient analysis of linear structural systems. The present algorithm employs twodecompositions. First, the total system is partitioned via spatial or domaindecomposition to obtain the governing equations of motions for each partitioneddomain. Second, for each partitioned subsystem, the governing equations aremodally decomposed into the rigid-body and deformational equations. Theresulting rigid-body equations are integrated by an explicit integrator, for itsstability is not affected by step-size restriction on account of zero frequencycontents (ω = 0). The modally decomposed partitioned deformation equations ofmotion are integrated by an unconditionally stable implicit integration algorithm.It is shown that the present AFETI-EI algorithm exhibits unconditional stabilityand that the resulting interface problem possesses the same solution matrix profileas the basic FETI static problems. The present simple dynamic algorithm, asexpected, falls short of the performance of the FETI-DP but offers a similarperformance of implicit two-level FETI-D algorithm with a much cheaper coarsesolver; hence, its simplicity may offer relatively easy means for conducting parallelanalysis of both static and dynamic problems by employing the same basicscalable FETI solver, especially for research-mode numerical experiments.
Acknowledgements: The first author has been partially funded by the localgovernment of Andalucıa (Junta de Andalucıa, Spain(P08-TEP-03804)) and theSpanish Ministry of Science (Ministerio de Educacion yCiencia(DPI2010-19331)). The second author has been partially supported byWCU (World Class University) Program through the Korea Science andEngineering Foundation funded by the Ministry of Education, Science andTechnology, Republic of Korea (Grant Number R31-2008-000-10045-0).
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FETI-Solvers for Non-standard Finite ElementEquations based on Boundary Integral OperatorsSession Schedule Author Index Session Index C18
Ulrich LangerJohann Radon Institute for Computational and Applied MathematicsAustrian Academy of Sciences, Linz, [email protected] HofreitherDoctoral Program “Computational Mathematics”Johannes Kepler University, Linz, [email protected] PechsteinInstitute for Computational MathematicsJohannes Kepler University, Linz, [email protected]
AbstractWe present efficient Domain Decomposition solvers for a class of non-standardFinite Element Methods. These methods utilize PDE-harmonic trial functions inevery element of a polyhedral mesh, and use boundary element techniques locallyin order to assemble the finite element stiffness matrices. For these reasons, theterms BEM-based FEM or Trefftz-FEM are sometimes used. In the present talk,we show that Finite Element Tearing and Interconnecting (FETI) metho ds canbe used to solve the resulting linear systems in a quasi-optimal and parallelmanner. An important theoretical tool are spectral equivalences between FEM-and BEM-approximated Steklov-Poincare operators.
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Contributed Talks C19Multiprocessors ApplicationsSchedule Author Index Session Index
Date: Tuesday, June 26Time: 14:45-15:35Location: PetriChairman: Eric Darrigrand
14:45-15:10 : Hatem LtaiefData-Driven Fast Multipole Method on Distributed Memory Systems withHardware AcceleratorsAbstract
15:10-15:35 : Menno GensebergerImproved Parallel Performance on Supercomputers by DomainDecomposition in Jacobi-Davidson for Large Scale Eigenvalue ProblemsAbstract
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Data-Driven Fast Multipole Method on DistributedMemory Systems with Hardware AcceleratorsSession Schedule Author Index Session Index C19
Hatem LtaiefSupercomputing Center, [email protected] YokotaCenter of Extreme Computing, [email protected]
AbstractFast N -body methods such as the fast multipole method (FMM) are essential forsolving particle-based systems and boundary integral problems with atwo-dimensional domain decomposition, because they can reduce the originalcomplexity of O(N2) to O(N). Besides the linear complexity, the FMM also hasfavorable characteristics such as; computationally intensive kernels, very fewsynchronization points, no iteration loops, and a hierarchical data structure thatadopts to the memory hierarchy. However, when the application has an irregulardistribution of particles/points the tree structure used in the FMM also becomesirregular, and simultaneous balancing of the work load load becomes a challengingproblem.Conventional techniques to handle the load-balancing problem in FMMs are basedon bulk-synchronous static repartitioning of the next step by using the work loadof the current step. The repartitioning can be done efficiently by updating the treestructure instead of rebuilding it, but the bulk-synchronous and static nature ofthe repartitioning poses certain limitations.The present work proposes an alternative approach based on data flowprogramming model, where data-driven dynamic scheduling is employed tobalance the work load within each subdomain. Previous work [1] by the authorsshowed promising results on shared memory systems. The idea is now to createper subdomain an instance of a dynamic runtime environment system (e.g.,StarPU [2]) and to dynamically schedule the different computational tasks on theavailable x86 CPU cores as well as GPUs, taking advantage of all processing unitsprovided by the system. The overall distributed FMM code can be therefore seenas a hybrid application in terms of scheduling policy, where a static schedulerdrives the application at the MPI level and a dynamic scheduler balances the workload at the level of the local computational domains.
[1] H. Ltaief and R. Yokota. Data-Driven Execution of Fast Multipole Methods.http://arxiv.org/abs/1203.0889, Submitted to EuroPar’12, August 2012.
[2] C. Augonnet, S. Thibault, R. Namyst, and P.-A. Wacrenier. StarPU: a unifiedplatform for task scheduling on heterogeneous multicore architectures. Concurr.Comput. : Pract. Exper., 23:187–198, February 2011.
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Improved Parallel Performance on Supercomputers byDomain Decomposition in Jacobi-Davidson for LargeScale Eigenvalue ProblemsSession Schedule Author Index Session Index C19
Menno [email protected]
AbstractMost computational work in Jacobi-Davidson, an iterative method for large scaleeigenvalue problems, is due to a so-called correction equation. In earlier work astrategy for the computation of (approximate) solutions of the correction equationwas proposed. The strategy is based on a domain decomposition preconditioningtechnique in order to reduce wall clock time and local memory requirements.However, there is more to gain. This talk discusses the aspect that the originalstrategy can be improved by taking into account that, for approximate solves ofthe correction equation by a preconditioned Krylov method, Jacobi-Davidsonconsists of two nested iterative solvers. For ease of presentation, consider thestandard eigenvalue problem A x = λx with an approximate eigenvalue θ,computed by Jacobi-Davidson so far, and preconditioner M ≈ A− θ I. In theinnerloop of Jacobi-Davidson a search subspace for the (approximate) solution ofthe correction equation is built up by powers of M−1 ( A− θ I ) for fixed θ. In theouterloop a search subspace for the (approximate) solution of the eigenvalueproblem is built up by powers of M−1 ( A− θ I ) for variable θ. In the domaindecomposition preconditioning technique was applied to the innerloop. But, as θvaries slightly in succeeding outer iterations, one may take advantage of thenesting by applying the same technique to the outerloop. For large scaleeigenvalue problems this aspect turns out to be nontrivial. In the talk, the impacton the parallel performance will be shown by results of scaling experiments onsupercomputers. This is of interest for large scale eigenvalue problems that need amassively parallel treatment.
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Contributed Talks C20Adaptive Meshing ParadigmSchedule Author Index Session Index
Date: Tuesday, June 26Time: 14:45-15:35Location: I50Chairman: Stephane Lanteri
14:45-15:10 : Shuo ZhangNorms of Trace Functions on Unstructured GridAbstract
15:10-15:35 : Cedric LachatPaMPA: Parallel Mesh Partitioning and AdaptationAbstract
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Norms of Trace Functions on Unstructured GridSession Schedule Author Index Session Index C20
Shuo ZhangLSEC, Institute of Computational Mathematics and Scientific/EngineeringComputing, Academy of Mathematics and System Sciences, Chinese Academy ofSciences, Beijing 100190, People’s Republic of [email protected] XuCenter for Computational Mathematics and Applications, Department ofMathematics, The Pennsylvania State University, University Park, [email protected]
AbstractNorm is an important concept that plays a key role in the design and analysis ofnumerical methods. Constructive or computational presentations of theappropriately chosen norms of discrete functions can bring much convenience intheory and practice. These presentations are not always available for the normsdefined non-locally and especially the dual norms. In this talk, we introduce theconstructive and computational presentations of several nonlocal norms of discretefunctions defined on the unstructured grid on the boundary of a domain. Theapproach is based on the construction of an isomorphic extension operator on thetrace space, and its conforming or nonconforming discretization. Discussions ofexactly revertible Poincare–Steklov operators are accompanied.
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PaMPA: Parallel Mesh Partitioning and AdaptationSession Schedule Author Index Session Index C20
Cedric LachatINRIA Sophia-Antipolis Mediterranee & INRIA Bordeaux [email protected] PellegriniUniversite Bordeaux 1 & INRIA Bordeaux [email protected] DobrzynskiIMB & INRIA Bordeaux [email protected]
AbstractThis talk will present the structure and operations of PaMPA (“Parallel MeshPartitioning and Adaptation”), a middleware library dedicated to the managementof unstructured meshes distributed across the processors of a parallel machine. Itspurpose is to relieve solver writers from the tedious and error prone task of writingagain and again service routines for mesh handling, data communication andexchange, remeshing, and data redistribution. PaMPA represents meshes asgraphs, whose data is distributed across the processors of the parallel machine.Graph vertices model the various entities of the mesh: its elements, faces, edges,nodes, etc. Edges connect interrelated entities: elements to all of their faces, facesto all of their nodes and edges, elements to their neighboring elements, etc.Numerical data of any type (either scalar, vector or structured) can be attached toeither kind of mesh entity or sub-entity. A mesh overlap size can be defined by theuser, so as to allow processors to access copies of mesh data located onneighboring processors. An overlap update routine allows users to propagatemodifications of the data associated with locally owned vertices to their copiesowned by neighboring processors. PaMPA iterators allow users to loop overentities and sub-entities of the mesh. By using iterators and accessing overlapdata, users can easily express their numerical schemes without having to writedata exchange routines by themselves. One of the key features of PaMPA is itsability to handle re-meshing in parallel. Parts of the mesh that need re-meshingare processed independently on each processor by a user-provides sequentialremesher. This processed is repeated on yet un-remeshed areas until all of themesh is processed. The re-meshed graph is then repartitioned so as to preserveload balance. In order to perform its task, PaMPA relies on several externallibraries. Parallel graph partitioning is performed by PT-Scotch, while sequentialremeshing of tetrahedra meshes is delegated to MMG3D.
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Contributed Talks C21FETI MethodsSchedule Author Index Session Index
Date: Tuesday, June 26Time: 14:45-15:35Location: I51Chairman: Alexandros Markopoulos
14:45-15:10 : Marta JarosovaHybrid Total FETIAbstract
15:10-15:35 : Michal MertaMassively Parallel Implementation of Total-FETI DDM with Applications toMedical Image RegistrationAbstract
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Hybrid Total FETISession Schedule Author Index Session Index C21
Marta JarosovaVSB-Technical University of Ostrava, Czech [email protected] MensıkVSB-Technical University of Ostrava, Czech [email protected] MarkopoulosVSB-Technical University of Ostrava, Czech [email protected] BrzobohatyVSB-Technical University of Ostrava, Czech [email protected]
AbstractWe propose a hybrid FETI method based on our variant of the FETI type domaindecomposition method called Total FETI. Our hybrid method was developed inan effort to overcome the bottleneck of classical FETI methods, namely the boundon the dimension of the coarse space due to memory requirements. We firstdecompose the domain into relatively large clusters that are completely separated,and then we decompose each cluster into smaller subdomains that are joinedpartly by Lagrange multipliers λ0 in selected interface variables or in averages ifthe transformation of basis is applied. The continuity in the rest of interfacevariables and also the Dirichlet condition are enforced by Lagrange multipliers λ1.This decomposition leads to the algorithm, where TFETI is used on two levels.The results of numerical experiments on benchmark from the linear elasticity willconclude the talk.
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Massively Parallel Implementation of Total-FETIDDM with Applications to Medical Image RegistrationSession Schedule Author Index Session Index C21
Michal MertaVSB-Technical University of [email protected] VasatovaVSB-Technical University of [email protected] HaplaVSB-Technical University of [email protected] HorakVSB-Technical University of [email protected]
AbstractThe FETI (Finite Element Tearing and Interconnecting) method turned out to beone of the most successful methods for the parallel solution of elliptic partialdifferential equations. The FETI-1 is based on the decomposition of the spatialdomain into non-overlapping subdomains that are “glued” by Lagrangemultipliers. Total-FETI (TFETI) by Dostal et al. simplifes the inversion ofstiffness matrices of subdomains by using Lagrange multipliers not only for gluingthe subdomains along the auxiliary interfaces, but also to enforce the Dirichletboundary conditions. Thus bases of kernels of all subdomain stiffness matrices areknown a priori and can be assembled directly from mesh data. In this work wecompare two parallel implementations of TFETI method based on either PETScor Trilinos software frameworks. Both these libraries are widely used for thedevelopement of scientific codes. While PETSc is based almost entirely on pure C,Trilinos utilizes features of the modern C++ including templates and objectoriented design. We focus on the parallel efficiency of both codes, mainly on thetreatment of the solution of the coarse problem and the action of orthogonalprojectors, which seem to be main bottlenecks of the TFETI parallelimplementations. Although usual applications of TFETI method lie in the field ofmaterial sciences and structural mechanics, we demonstrate applicability of ourcodes to the problem of the image registration of computer tomography andmagnetic resonance imaging data using elastic registration method. The numericalbenchmarks were run on HECToR supercomputer at EPCC in the UK which ispart of the PRACE HPC ecosystem.
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Author Index
Abgrall, Remi, 85Abhyankar, Shrirang, 132Acebrn, Juan A., 314Adler, James, 55Alart, Pierre, 261Alonso Rodriguez, Ana, 117Andra, Heiko, 166Andzembe Okoubi, Firmim, 313Anitescu, Mihai, 132Antoine, Xavier, 108Antonietti, Paola F., 230Apoung Kamga, Jean-Baptiste, 50Arbogast, Todd, 210Aristidou, Petros, 324Aubert, Stephane, 292Auliac, Sylvain, 201Ayache, Nicholas, 245Ayuso de Dios, Blanca, 28, 230, 233
Bansch, Eberhard, 26Belanger-Rioux, Rosalie, 115Badea, Lori, 332Badia, Santiago, 286Baffico, Leonardo, 291Barker, Andrew T., 231Baudron, Anne-Marie, 268Beirao da Veiga, Lourenco, 94, 133Belhachmi, Zakaria, 204Ben Belgacem, Faker, 201
Benabderrahmane, Benyattou, 255Bercovier, Michel, 86Berenguer, Laurent, 288Bernal, Francisco, 314Berninger, Heiko, 43, 73, 142, 147,
215Berthe, Paul-Marie, 49Bertolazzi, Enrico, 117Bertoluzza, Sivia, 40Bertrand, Fleurianne, 57Betcke, Timo, 220Betsch, Peter, 90Blanco, Pablo Javier, 74Blayo, Eric, 99, 277, 327Bochev, Pavel, 56Borsboom, Mart, 101Boubendir, Yassine, 108Brannick, James, 165Brenner, Susanne C., 228, 230, 231Brezzi, Franco, 233Brinkhues, Sarah, 151Brix, Kolja, 226Brzobohaty, Tomas, 209, 257, 358Buck, Marco, 166Buffa, Annalisa, 96
Cai, Xiao-Chuan, 24, 296Calo, Victor M., 87Campos Pinto, Martin, 226
361
Canuto, Claudio, 225, 226Casoni, Eva, 75Castro, Carlos, 75Cermak, Martin, 298Chabannes, Vincent, 39Chacon Rebollo, Tomas , 342Chacon Vera, Eliseo, 342Cheik Ahamed, Abal-Kassim, 269Chen, Zhangxin, 46Cheng, Yu-Fen, 285Cherel, David, 327Cho, Durkbin, 94, 133Choi, Daniel, 260Chung, Eric T., 129Ciarlet Jr., Patrick, 268Claeys, Xavier, 32Colli Franzone, Piero, 152Collier, Nathan, 87Conen, Lea, 105Contreras, Ivan, 182Cotin, Stephane, 245Cresta, Philippe, 262Croce, Roberto, 309Crosetto, Paolo, 154
Dahmen, Wolfgang, 226Dai, Xiaoying, 185Dalcin, Lisandro, 87Dao, Thu Huyen, 304Darbas, Marion, 112Dari, Enzo Alberto, 74Darrigrand, Eric, 112de la Cruz, Luis Miguel, 180Debreu, Laurent, 99Degond, Pierre, 145Delingette, Herve, 245Demanet, Laurent, 115Deparis, Simone, 154Desmeure, Geoffrey, 262Dickopf, Thomas, 238, 335Dimarco, Giacomo, 145Discacciati, Marco, 76, 280Dobrzynski, Cecile, 85, 355Dohrmann, Clark, 68, 135, 161Dolean, Victorita, 38, 107, 122, 123,
157, 158
Dostal, Zdenek, 209Dryja, Max, 127Du, Rui, 252Dufaud, Thomas, 174Dureisseix, David, 261Duriez, Christian, 245Dyyak, Ivan I., 256
Efendiev, Yalchin, 128, 160, 168Eguzkitza, Beatriz, 305, 343Emmett, Matthew, 186, 310Engquist, Bjorn, 121Engwer, Christian, 282Erhel, Jocelyne, 173, 308Erlangga, Yogi Ahmad, 113
Fabozzi, Davide, 324, 326Feng, Hui, 136Fernandez, Miguel, 153Fischle, Andreas, 151Flemisch, Bernd, 48Folzan, Gauthier, 334Fomel, Sergey, 121Frenod, Emmanuel, 141, 142, 147Froehly, Algiane, 85
Guttel, Stefan, 190Gahalaut, Krishan, 88, 95Gallimard, Laurent, 260Galvis, Juan, 128, 160, 168, 250Gander, Martin J., 61, 62, 67, 70,
106, 107, 122, 123, 142, 147,188, 190, 191, 195, 214, 218,220, 221, 320
Gdoura, Mohamed Khaled, 303Genseberger, Menno, 351Gerardo-Giorda, Luca, 237Gervasio, Paola, 76Geuzaine, Christophe, 108, 326Geymonat, Giuseppe, 281Ghiloni, Riccardo, 117Gibbon, Paul, 310Golse, Francois, 144Gonzalez, Jose A. , 346Gosselet, Pierre, 262, 321Gostaf, Kirill Pichon, 340
362
Graham, Ivan G., 164, 169Grandperrin, Gwenol, 154Greif, Chen, 284Guba, Oxana, 56Guetat, Rim, 187
Haeberlein, Florian, 51Hairer, Ernst, 191Hajian, Soheil, 218Halpern, Laurence, 8, 51, 67, 70, 216,
266Hapla, Vaclav, 359Harbrecht, Helmut, 96Hardt, Steffen, 146Haslinger, Jaroslav, 254Hauret, Patrice, 158Have, Pascal, 45Haynes, Ronald, 197, 219Hecht, Frederic, 37, 38, 201Helmig, Rainer, 48Hendili, Sofiane, 281Herrera, Ismael, 179, 181, 182Hesch, Christian, 90Hintermuller, Michael, 312Hiptmair, Ralf, 32Hoang, Thi Thao Phuong, 44Hofreither, Clemens, 347Hohage, Thorsten, 116Holst, Michael, 34Horak, David, 359Houzeaux, Guillaume, 305, 343Hsieh, Ben, 46Hu, Xiaozhe, 171Huang, Tsung-Ming, 285Huber, Martin, 118Hubert, Florence , 214, 266Hwang, Feng-Nan, 285
Iliev, Oleg, 166, 290Ipopa, Mohamed Ali, 303
Juttler, Bert, 91Jerusalem, Antoine, 343Jaffre, Jerome, 44Jamelot, Erell, 268Japhet, Caroline, 44, 49, 203
Jarosova, Marta, 358Jerez-Hanckes, Carlos, 32Jiang, Daijun, 136Jimenez-Perez, Hugo, 192Jolivet, Pierre, 38Julien, Salomon, 196
Kammogne, Rodrigue, 325Kang, Kab Seok, 330Kanschat, Guido, 232Karamian, Philippe, 264Karangelis, Anastasios, 213Keilegavlen, Eirik, 170Kern, Michel, 44Keyes, David, 296Kim, Hyea Hyun, 20, 129, 130Klar, Axel , 146Klawonn, Axel, 12, 151Kleiss, Stefan, 91Koko, Jonas, 303, 313Kornhuber, Ralf, 43, 73, 276Kovar, Petr, 273Kozubek, Tomas, 257Krasucki, Francoise, 281Kraus, Johannes, 88, 95, 175, 176Krause, Dorian, 238Krause, Rolf, 105, 238, 309, 310, 335Krell, Stella, 214Krishnan, Abilash, 337Krzyzanowski, Piotr, 127Kumar, Pawan, 331Kunoth, Angela, 93, 96Kucera, Radek, 254Kwok, Felix, 188, 195, 297
Laadhari, Aymen, 244Laayouni, Lahcen, 267Lachat, Cedric, 355Lafranche, Yvon, 112Langer, Andreas, 312Langer, Ulrich, 347Lanteri, Stephane, 120, 122, 123Laskar, Jacques, 192Lautard, Jean-Jacques, 198, 268Le Tallec, Patrick, 139, 334Leclerc, Willy, 264
363
Lee, Chang-Ock, 20, 130Lee, Jin-Fa, 16, 111, 122Lee, Jungho, 132Lehrenfeld, Christoph, 227Lemarie, Florian, 99Lemou, Mohammed, 140Leskinen, Jyri, 302Li, Jing, 134Li, Liang, 120Li, Siwei, 121Linel, Patrice, 299Liu, Hui, 46Loghin, Daniel, 293, 315, 325Loisel, Sebastien, 213, 215Ltaief, Hatem, 350Lukas, Dalibor, 273Lymbery, Maria, 175
Munzenmaier, Steffen, 58Maday, Yvon, 185, 187, 196, 198,
203, 207Magoules, Frederic, 187, 269, 304,
305, 337Makhoul-Karam, Noha, 308Maly, Lukas, 273Mandal, Bankim Chandra, 188, 195Mao, Youli, 232Marchesseau, Stephanie, 245Marcinkowski, Leszek, 318Margenov, Svetozar, 175Marini, L. Donatella, 233Markopoulos, Alexandros, 257, 298,
358Martın, Alberto, 286Martin, Alexandre, 261Martynyak, Rostyslav M., 256Masson, Roland, 45Maynard, Chris, 213McInnes, Lois, 132Melenk, Jens Markus, 206Mensık, Martin, 358Merta, Michal, 359Michaud, Jerome, 142, 147Michel, Anthony, 51Minion, Michael, 186, 310Mirabella, Lucia, 237
Mula Hernandez, Olga, 198Munson, Todd, 132
Nannen, Lothar, 116Nassif, Nabil, 308Nataf, Frederic, 38, 45, 157, 158, 203Ndjinga, Michael, 304Nordbotten, Jan Martin , 170Notay, Yvan, 331Nouiri, Brahim, 255
Ohlberger, Mario, 73Omnes, Pascal, 49Ong, Benjamin, 197
Periaux, Jacques, 302Pacull, Francois, 292Pardo, David, 87Park, Eun-Hee, 20, 228, 231Park, K. C., 346Parret-Freaud, Augustin, 321Paszynski, Maciej, 87Patel, Ajit, 341Pavarino, Luca Franco, 94, 133, 152,
225Pechstein, Clemens, 18, 91, 158, 161,
347Pellegrini, Francois, 355Peng, Zhen, 111, 122Pennacchio, Micol, 40Perego, Mauro, 237Perlat, Jean-Philippe, 334Pernice, Michael, 296Perotto, Simona, 78Perrussel, Ronan, 120, 123Phillips, Joel, 220Picard, Christophe, 39Pichot, Geraldine, 10Pieri, Alexandre B., 225Pierre, Charles, 240Pironneau, Olivier, 340Plank, Gernot, 243Plumier, Frederic, 326Poirriez, Baptiste, 173Potse, Mark, 238Poulson, Jack, 121
364
Pradhan, Debasish, 336Prignitz, Rodolphe, 26Principe, Javier, 286Prokopyshyn, Ihor I., 256Prokopyshyn, Ivan A., 256Prud’homme, Christophe, 39, 40
Quarteroni, Alfio, 76, 154, 202, 244
Rahman, Talal, 249, 318Rammerstorfer, Franz, 79Rey, Christian, 262, 321Rey, Valentine, 321Rezaijafari, Human, 80Rheinbach, Oliver, 151Riahi, Mohamed Kamel, 193, 196Riaz, Samia, 315Riton, Julien, 263Riviere, Beatrice, 22Roberts, Jean, 44Rosas, Alberto, 181Rousseau, Antoine, 277, 327Roux, Francois-Xavier, 63, 208, 319,
340Ruiz Baier, Ricardo, 244Ruprecht, Daniel, 309, 310
Salomon, Julien, 193Samake, Abdoulaye, 39, 40Sander, Oliver, 43, 73, 81, 215, 276Sandve, Tor Harald, 170Sangalli, Giancarlo, 96Santugini, Kevin, 67, 70Sarkis, Marcus, 14, 127Sassi, Taoufik, 254, 260, 263, 291,
303Scacchi, Simone, 94, 133, 152Schadle, Achim, 116Schoberl, Joachim, 116, 118, 227Schanz, Martin, 79Scheichl, Robert, 157, 158, 164, 251Schiela, Anton, 81Schroder, Jorg, 151Sermesant, Maxime, 245Shao, Lei, 46Smetana, Kathrin, 73
Smith, Barry, 132Soloveichik, Ilya, 86Speck, Robert, 310Spillane, Nicole, 38, 157, 158Starke, Gerhard, 57, 58Steinbach, Olaf, 110Stolk, Chris, 272Sugny, Dominique, 193Sung, Li-yeng, 228, 230, 231Sysala, Stanislav, 298Szydlarski, Mikolaj, 45Szyld, Daniel, 267, 284
Talbot, Hugo, 245Tayachi, Manel, 277Thierry, Bertrand, 108Thies, Jonas, 102Thomines, Florian, 168Tiwari, Sudarshan, 146Tjahjanto, Denny, 343Tomar, Satyendra, 88, 91, 95Toulougoussou, Ange, 319Tran, Minh-Binh, 216Tromeur-Dervout, Damien, 288, 299Tu, Xuemin, 62, 134Tyrone, Rees, 284
Vazquez, Mariano, 343Vazquez, Rafael, 96Valli, Alberto, 117van ’t Hof, Bas, 100Van Cutsem, Thierry, 324, 326Vanek, Petr, 163Vasatova, Alena, 359Vassilevski, Panayot S., 55, 160, 251Veneros, Erwin, 107Veneziani, Alessandro, 237Vidrascu, Marina, 281Vion, Alexandre, 108Visseq, Vincent, 261Vlach, Oldrich, 209
Waluga, Christian, 206Wang, Lei, 132Wang, Weichung, 285Weiser, Martin, 242
365
Westerheide, Sebastian, 282Widlund, Olof B., 64, 68, 129, 135Willems, Joerg, 69Winkel, Mathias, 310Wohlmuth, Barbara, 80, 206Wubs, Fred, 102
Xiao, Hailong, 210Xu, Jinchao, 159, 233, 354Xu, Yingxiang, 221
Yang, Chao, 296Ying, Lexing, 121Yokota, Rio, 350Youett, Jonathan, 276Yu, Song, 46
Zampini, Stefano, 239Zemitis, Aivars, 290Zemzemi, Nejib, 153Zhang, Chen-Song, 30Zhang, Hui, 106, 320Zhang, Shuo, 354Zhao, Tao, 45Zikatanov, Ludmil, 164, 233, 251Zou, Jun, 136
366
