Multiscale Basis Functions for Iterative Domain ... · Motivation: Domain Decomposition Method The...

Motivation: Domain Decomposition MethodThe Multiscale Mixed Method (MuMM)

Numerical ResultsConclusions and Future Work

Multiscale Basis Functions for Iterative Domain DecompositionProcedures

A. Francisco1, V. Ginting2, F. Pereira3 and J. Rigelo2

1Department Mechanical EngineeringFederal Fluminense University, Volta Redonda, RJ 27255-125, Brazil

[email protected]!.br

2Department of MathematicsUniversity of Wyoming, Laramie, WY 82071-3036, USA

{vginting,jrigelo}@uwyo.edu

3Department of Mathematics and School of Energy ResourcesUniversity of Wyoming, Laramie, WY 82071-3036, USA

[email protected]

Support: DOE: DE-FE0004832/DE-SC0004982; NSF: DMS-1016283;Center for Fundamentals of Subsurface Flow(UW).



Outline:

1 Motivation: Domain Decomposition Method

2 The Multiscale Mixed Method (MuMM)

3 Numerical Results

4 Conclusions and Future Work



Motivation

We are concerned withthe development ofnumerical proceduresfor the fast andaccurate approximationof subsurface flows thatcan take advantage ofheterogeneousprocessing units.



Motivation

Incorporate fine scale information into a coarse scalediscretization, without solving it directly.

Coarse Domain Decomposition

Our iterative procedure does not use MPI in each iteration.



Model Problem

Our model problem is a second order linear elliptic equationwritten as a first order system

!.u = f (x), where u = "k(x)!p in !, (1)

p = pb on "D , u.! = ub on "N . (2)

Here ! is a bounded domain with a Lipschitz boundary"! = "D # "N , "D $ "N = %.



Domain Decomposition

The domain ! is divided into a non-overlapping partition{!j}:

! =!M

j=1!j ; !j $ !k = %, j &= k .

Motivation: Non-overlapping iterative DDM based on theRobin boundary conditon.

J. Douglas, Jr., P. J. Paes Leme, J. E. Roberts, and J. Wang,A parallel iterative procedure applicable to the approximatesolution of second order partial di!erential equations by mixedfinite element methods, Numer. Math., 65 (1993) 95–108.



Domain Decomposition (cont.)

Mixed finite element space: hybridized Raviart-Thomas.

Procedure: subdomain = element.

Degrees of freedom (for each !j) :

p, u! and #! , $ = L,R ,B ,T .h

B

T

L R

Then for a single element, the discrete form of the Poisson’sequation is given by

uL + uR + uB + uT = fh, (3)

u! " 2

hk(p " #!) = 0. (4)

(5)




Robin Interface Condition:

#! = %!(u! + u!!) + #!! , where $ = L,R ,B ,T . (6)

!j !k $ - $!

L

T

R

T

B

R

L

B

"jk




Douglas, Jr. et al. parallel iterative scheme:

1 Set an initial guess: {p0, u0! , #0!}.2 For all red elements, update {p, u! , #!}, using [3, 4, 5].

3 For all black elements, compute {p, u! , #!}, by solving[3, 4, 5], using the updated values from the red elements.

4 Check for convergence.

new

old

old

oldold




o

o o

o

n

o

o o

o

n

subdomain: one element larger subdomain

Convergence is established.



The Multiscale Basis Functions Formulation

Consider a subdomain !j . Let &ji = (ui! , #i! , p

i )j , i = 1, ..., 4N,be the basis functions associated with this subdomain.

"%LuL+#L = 1 '0

0 0 0 0

0

0

0

0

0000

0

0

&j1

B. Ganis and I. Yotov, Implementation of a mortar mixed finiteelement method using a multiscale flux basis, Computer Methods inApplied Mechanics and Engineering, 198 (2009) 3989-3998.




Given the Robin boundary values Aji , the solution for thePoisson equation is given by

S"j =4N"

i=1

Aji&ji j3

N

N

A

Aj1

j2A

where, for i = 1, ..., 4N,&ji = (ui! , #

i! , p

i )j are the canonical basis functions.




Advantage :

Avoid the direct solution of the local problems.

Problem :

We have to compute 4N basis functions for each subdomain!



MuMM: A Modified iteration

Introduce an intermediate scale H, h ( H ( H.

Based on an average Robin condition:

Aji = "%LuTL +uBL

2 +"TL +"BL

2h

TB

Aji H

H

!j !k

Goal: To reduce the number of basis functions.




The solution is given by, for example: S"j =

4N/2"

i=1

Aji &ji .

"%LuL+#L = 1 '

0

0 0

0 0

0

0

&j1

A

N

Aj2

j1

2D: Douglas, Jr. et al. iteration; 3D: CG preconditioned withthe AMG.

Solution in the fine grid: post-processing.




Remarks :

Flux conservation is maintained in the H scale.

The balance between numerical accuracy and numericale#ciency is determined by the choice of

span{&ji} ) span{&ji}.

Extreme cases:

H = h: Douglas, Jr. et al. iteration.H = H: 4 basis functions/subdomain.



Example 1: kmax/kmin = 176

Example: 2D problem with a fine grid of 220* 60,coarse grid of 11* 3 (subdomains of 20* 20).

Permeability model: SPE10 model, where k(x) = exp(' ((x)).

20 40 60 80 100 120 140 160 180 200 220

5

10

15

20

25

30

35

40

45

50

55

60

5

10

15

20

25

The physical transport of fluids is given by solving:

)#c#t + u.!c = 0, with I .C .+ B .C . given.



H = H

H = H/4

H = H/2

fine grid



Tracer cut curves

The fraction of the tracer in the produced fluid is given by

F(t) =

#!"out

c u.n dS#!"out

u.n dS.



Relative Errors :

Relative error = !uMuMM"ufine!maxi,j!ufine! .

Figure :From top to bottom:

4, 8 and 16 basis functions.



Example 2: kmax/kmin = 1408.

Example: 2D problem with a fine grid of 220* 60,coarse grid of 11* 3 (subdomains of 20* 20).

Permeability model: SPE10 model. We consider 16 basisfunctions.

MuMM fine grid



Example 2: Tracer cut curve and permeability field



Conclusions

Properties :

uL + uR + uB + uT = fh holds in the fine grid.

Sources and sinks are naturally incorporated in the procedure.

All local problems are positive definite.

Global information is not needed.

Straightforward implementation in 2 and 3D.

Fits well in CPU-GPU clusters.



Future Work

3D implementation on GPUs.

Extension to multiphase/Compositional flows.

Adaptivity (basis functions not altered).

Enrichment of basis functions.

Thank you!!



References

J. Douglas, Jr., P. J. Paes Leme, J. E. Roberts, and J. Wang,A parallel iterative procedure applicable to the approximatesolution of second order partial di!erential equations by mixedfinite element methods, Numer. Math., 65 (1993) 95–108.

B. Ganis and I. Yotov, Implementation of a mortar mixed finiteelement method using a multiscale flux basis, ComputerMethods in Applied Mechanics and Engineering, 198 (2009)3989-3998.

Vegard Kippe . Jorg E. Aarnes. Knut-Andreas Lie, Acomparison of multiscale methods for elliptic problems inporous media flow, Comput Geosci, (2008) 12:377-398.



Variational Formulation

The pressure and velocity spaces for the global problem [1, 2] are:

W = L2(!) and Vr = {v + H(div ;!)| v.! = r on "N},

where H(div ;!) = {v + (L2(!))2| div v + L2(!)}.

The global weak form is giving by finding {p,u} + W * Vr suchthat

(K"1u, u)" " (p, div u)" = 0, u + V0, (7)

(div u, p)" = (f , p)", p + W . (8)



Variational Formulation

Similarly, define the spaces for each subdomain !j by

Wj = {w |!j | w + W (!)},

Vr ,j = {v + H(div ;!j) | v.!j = r on "!j $ "N}.

The weak formulation are given by seeking {pj ,uj} + Wj * Vr ,j

such that

(div u, p)"j = (f , p)"j , p + Wj ,

(K"1u, u)"j " (p, div u)"j +"

j #=k

< p, u.!j >#jk=

"M"

j

< pb, u.!j >#"j$#D , u + V0,j ,

where "jk = "kj = "!j $ "!k .

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