Adaptive finite elements with large aspect ratio : theory ...sma.epfl.ch/~picasso/crete.pdf ·...

Adaptive finite elements with large aspect ratio :theory and practice

M. Picasso

Institut d’Analyse et Calcul ScientifiqueEcole Polytechnique Federale de Lausanne

Switzerland

Crete 2008

M. Picasso Adaptive finite elements with large aspect ratio

Example 1 : 2D advection-diffusion

Mesh Isovalues

Boundary layer 10−4, 241 vertices, asp. ratio O(105), sameprecision with isotropic mesh O(105) vertices, SISC 2003.

Design of the stabilization parameter : anisotropic a priorierror estimates, with S. Micheletti and S. Perotto, SINUM2003.


Example 2 : 3D dendritic growth

Parabolic nonlinear system, existence E. Burman and J.Rappaz, M2AS 2003.Anisotropic a posteriori error estimates, with E. Burman,Interfaces Free Bound. 2003.Dendritic growth with convection, with J. Narski, CMAME2007, Fluid Dyn. Mater. Proc. 2007.Animation MeshAdapt mesh generator, INRIA Gammaproject, Distene company.


Example 3 : 3D supersonic flow around an aircraft

With Y. Bourgault (Ottawa), collaboration with INRIAGamma project, F. Alauzet, A. Loseille, supported by DassaultAviation, IJNMF 2008.

Compressible Euler solver (F. Alauzet), Anisotropic meshgenerator (C. Dobrzynski, P. Frey), Anisotropic errorestimator.

Animation, 106 vertices, single 32b processor.


Example 4 : Microfluidics

With V. Prachittham, unsteady convection-diffusion flows.

Space-time adaptive finite element with large aspect ratio.

Optimal a posteriori error estimates for the Crank Nicolsonscheme.

Akrivis Nochetto Makridakis, Math. Comp. 2006.

Animation.

Animation.


Outline

The anisotropic error estimator for the Laplace problem.

The anisotropic error estimator for the heat problem with theCrank Nicolson scheme.


The anisotropic error estimator for the Laplace problem

Find u : Ω → R such that

−∆u = f in Ω,

u = 0 on ∂Ω.

Let Th be a mesh of Ω into triangles K with diameter hK lessthan h.

Find uh ∈ Vh (continuous, piecewise linears) such that, for allvh ∈ Vh ∫

Ω∇uh · ∇vh =

∫Ω

fvh.


Anisotropic interpolation estimates

Formaggia Perotto, Numer. Math. 2001, 2003.

See also Kunert Verfurth, Numer. Math. 2000.

x1

x2

x1

x2

1

1

H

h

r1,K

r2,K

TK

K K

x = TK (x) = MK x + tK , s. v. d. MK = RTK ΛKPK

RK =

(rT1,K

rT2,K

)=

(1 00 1

), ΛK =

(λ1,K 0

0 λ2,K

)=

(H 00 h

).





x1

x2 TK

1K

K

r1,K

r2,K

λ1,K

λ2,K

x = TK (x) = MK x + tK , s. v. d. MK = RTK ΛKPK





The number of neighbours must be bounded aboveindependently of h.

There is a (technical) restriction of the reference patchT−1

K (∆K ) which must be O(1). This excludes meshes having“large curvature”. In practice, anisotropic mesh generatorsexclude such meshes.


The anisotropic a posteriori error estimator

Upper bound : ∃C > 0 indep. mesh size and aspect ratio s.t.∫Ω

|∇(u − uh)|2dx

≤ C∑K∈Th

((∫K

f 2

)1/2

+1

2

(|∂K |

λ1,Kλ2,K

)1/2(∫∂K

[∇uh · n]2)1/2

)(

λ21,K

(rT1,KGK (u − uh)r1,K

)+ λ2

2,K


))1/2

,

with GK (u − uh) = GK (e) =

∫

K

(∂e

∂x1

)2 ∫K

∂e

∂x1

∂e

∂x2∫K

∂e

∂x1

∂e

∂x2

∫K

(∂e

∂x2

)2

.

Lower bound if the mesh equidistributes the error in the directionsof maximum and minimum stretching :

λ21,K


)≤ λ2

2,K


)∀K ∈ Th.


The anisotropic a posteriori error estimator

The error estimator (e = u − uh) :

η2K =

(‖f ‖L2(K) +

1

2

(|∂K |

λ1,Kλ2,K

)1/2

‖ [∇uh · n] ‖L2(∂K)

)(

λ21,K

(rT1,KGK (e)r1,K

)+ λ2

2,K

(rT2,KGK (e)r2,K

))1/2

.

How to approach GK (e) =

∫

K

(∂e

∂x1

)2 ∫K

∂e

∂x1

∂e

∂x2∫K

∂e

∂x1

∂e

∂x2

∫K

(∂e

∂x2

)2

?

Zienkiewicz-Zhu (Πh is an approximate L2 projection onto Vh)∫K

(∂e

∂x1

)2

=

∫K

(∂(u − uh)

∂x1

)2

→∫

K

(∂uh

∂x1− Πh

∂uh

∂x1

)2

.


Anisotropic, adaptive finite elements

Goal : find Th s.t. 0.75 TOL ≤

∑K∈Th

η2K

1/2

(∫Ω |∇uh|2

)1/2≤ 1.25 TOL

Sufficient condition (NK is the number of triangles) :0.752TOL2

∫Ω |∇uh|2

NK≤ η2

K ≤1.252TOL2

∫Ω |∇uh|2

NKSufficient condition in the directions of maximum (i = 1) andminimum (i = 2) stretching :

0.752TOL2∫Ω|∇uh|2√

2NK

≤

(‖f ‖L2(K) +

1

2

(|∂K |

λ1,Kλ2,K

)1/2

‖ [∇uh · n] ‖L2(∂K)

)(λ2

i,K

(rTi,KGK (e)ri,K

))1/2

≤1.252TOL2

∫Ω|∇uh|2√

2NK


Anisotropic, adaptive finite elements

P

h1,P

h2,P

θP

Equidistribute the error in directions 1 and 2

Align the triangle with the eigenvectors of GK (e)

2D : use the BL2D mesh generator (INRIA, Borouchaki, Laug)

3D : use the MeshAdapt mesh generator (INRIA, George,Distene) or the mmg3d mesh generator (Dobrzynski, Frey).


Adaptive meshes for the Laplace problem in 2D

Initial 10× 10 mesh


Adaptive meshes for the Laplace problem in 2D

TOL = 0.25 : adapted mesh after 30 mesh generations, 145 vertices


Effectivity index with anisotropic adapted meshes

TOL vertices error ei eiZZ asp. ratio

0.125 854 0.25 2.70 1.00 2620.0625 2793 0.13 2.75 0.99 2880.03125 10812 0.062 2.79 0.95 4250.015625 42562 0.031 2.79 0.98 1199

The effectivity index is aspect ratio independent on adaptedmeshes


The heat equation with Crank-Nicolson scheme

Optimal a posteriori error estimates for Crank-Nicolson timediscretization : Akrivis Makridadkis Nochetto, Math. Comp.2006.

With V. Prachittham and A. Lozinski :∂u

∂t−∆u = f , time

and space discretization.

For n = 1, ...,N, find unh ∈ Vh such that, for all vh ∈ Vh

1

τn

∫Ω(un

h−un−1h )vdx+

1

2

∫Ω∇(un−1

h +unh)·∇vdx =

1

2

∫Ω(f n−1+f n)vdx .

uhτ (x , t) =t − tn−1

τnunh(x) +

tn − t

τnun−1h (x).∫

Ω

∂uhτ

∂tvhdx +

∫Ω∇uhτ · ∇vhdx

=1

2

∫Ω(f n−1 + f n)vhdx + (t − tn−1/2)

∫Ω∇∂uhτ

∂t· ∇vhdx .


Two possible time reconstructions

The Two-Point Time Reconstruction : following AkrivisMakridadkis Nochetto, Math. Comp. 2006

uhτ (x , t) = uhτ (x , t)+1

2(t−tn−1)(t−tn)wn

h , t ∈ In, n = 1 · · ·N,

where wnh ∈ Vh is defined by∫

Ωwn

h vh dx =

∫Ω

f n − f n−1

τn−∫

Ω∇∂uhτ

∂t· ∇vh dx , ∀vh ∈ Vh.


Two possible time reconstructions

The Three-Point Time Reconstruction : for tn−1 ≤ t ≤ tn

uhτ (x , t) = uhτ (x , t) +1

2(t − tn−1)(t − tn)∂2

nuh,

where

∂2nuh =

unh − un−1

h

τn−

un−1h − un−2

h

τn−1

(τn + τn−1)/2.

or equivalently

uhτ (x , t) =(t − tn−2)(t − tn−1)

(tn − tn−2)(tn − tn−1)unh

+(t − tn)(t − tn−2)

(tn−1 − tn)(tn−1 − tn−2)un−1h

+(t − tn)(t − tn−1)

(tn−2 − tn)(tn−2 − tn−1)un−2h .


Error indicator

Approach

∫ T

t1

∫Ω

|∇(u − uhτ )|2 byN∑

n=2

∑K∈Th

η2K ,n with η2

K ,n =

∫ tn

tn−1

(‖f − ∂nuh + ∆uhτ‖L2(K) +

1

2λ1/22,K

‖[∇uhτ · n]‖L2(∂K)

)(

λ21,K

(rT1,KGK (u − uhτ )r1,K

)+ λ2

2,K

(rT2,KGK (u − uhτ )r2,K

))1/2

+

∥∥∥∥f − (f n−1/2 + (t − tn−1/2)f n − f n−2

τn + τn−1

)∥∥∥∥2

L2(K)

+τ 4n

∥∥∇∂2nuh

∥∥2

L2(K)

dt.


Adaptive space-time algorithm

Choose the time step and the mesh size so that

0.875TOL ≤ ηspace + ηtime(∫ T

0

∫Ω|∇uhτ |2 dx dt

)1/2≤ 1.125TOL.

Test case 1 Animation

Test case 2 Animation

TOL error eiZZ ei space ei time Vert N Mes. AR0.125 0.03 0.99 2.87 2.68 155 142 84 630.0625 0.015 0.99 2.89 2.91 348 201 52 1080.03125 0.0078 0.99 2.96 2.99 892 285 52 1650.015625 0.0040 1.00 2.88 2.71 4408 401 40 118

Conclusion : Vert = O(TOL−2) and N = O(TOL−1/2).


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