homogenization finite elements · impa versieux & sarkis 1u0 h1 0 h2 ∂u0 ∂x n s 0 and ∂u0...

HOMOGENIZATION FINITE ELEMENTS

Henrique M. Versieux

([email protected])

and

Marcus Sarkis

([email protected])

Instituto Nacional de Matematica Pura e Aplicada (IMPA)

Rio de Janeiro, Brazil

1

Outline

IMPA Versieux & Sarkis� Setting out the Problem

� Asymptotic Expansion

� Theoretical Boundary Corrector Approximation

� Numerical Method

� Convergence Results

� Numerical Experiments

� Conclusions

2

Setting out the Problem

IMPA Versieux & Sarkis

We are interested in a numerical method for solving

Lεuε� � ∂∂xi

ai j � x � ε �

∂∂x j

uε� f in � , uε� 0 on ∂ �

� a � y � � � ai j � y � � is symmetric positive definite

� ai j is periodic with periodic cell named Y

� ai j � C1,β , β � 0

� � � Y� � 0, 1 � 0, 1

� The size of the mesh used is h � ε � or h � � ε �

3

Related Works

IMPA Versieux & Sarkis� I. Babuska (1975)

� T. Hou et al (1997), (1999), (2000), and (2004)

� Y. Efendiev and X. H. Wu (2001)

� Weinan E and B. Engquist (2003)

� G. Sangalli (2003)

� G. Allaire and R. Brizzi (2004)

� C. Schwab, A. M. Matache (2002)

4

Asymptotic Expansion


� Asymptotic expansion ansatz

uε � x � � u0 � x, x � ε � � εu1 � x, x � ε � � ε2u2 � x, x � ε � � � � �

where the functions u j � x, y � are Y periodic in y

� We consider a first order approximation plus a boundary

corrector term

� Let χ j � H1 � Y � be the Y periodic solution with zero average on

Y of

� y

� a � y � � yχj� � y

� a � y � � y y j�

∂∂yi

ai j � y �

� Define the symmetric positive definite matrix

Ai j�

1

�Y � Yalm � y �

∂ � χi � yi �

∂yl

� ∂ � χ j � y j �

∂ymdy

5

Asymptotic Expansion


� Define u0 � H1 � � � as the solution of

� � .A � u0 � f in � u0� 0 on ∂ �

� Let

u1 � x,x

ε �� χ j

�

x

ε �

∂u0

∂x j

� x �

� θε � H1 � � � is introduced as the solution of

� � � a � x � ε � � θε� 0 in � ; θε� � u1 � x,x

ε � on ∂ �

Proposition 1 (Moscow/Vogelius (1997)). There exists a constant c

independent of u0 and ε, such that:

� uε � � � � u0 � � � � εu1 � � , � � ε � � εθε � � � � 1 � cε � u0 � 2

� uε � � � � u0 � � � � εu1 � � , � � ε � � εθε � � � � 0 � cε2� u0 � 3

6

Boundary Corrector Approximation


� The equation defining θε is as hard to solve as the original

problem

Lεθε � 0 on � , θε � � u1 � x, x � ε � on ∂ � .

� To approximate θε we decompose θε � θε � θε

Lεθε � 0 in � , θε� � u1

� χ� ∂ηu0 on ∂ �

Lεθε � 0 in � , θε � χ � ∂ηu0 on ∂ �

χ � �

��

��

�

χ �e on � e

χ �w on � w

χ �n on � n

χ �s on � s,

� � e� w� n

� s

7



� u0 � H10 � � � � H2 � � � � ∂u0

∂x1 � � n � � s

� 0 and ∂u0

∂x2 � � e� � w

� 0

� ∂ηu0 � � k � H1 � 200 � � k � � χ� ∂ηu0 � H1 � 2 � ∂ � �

� Approximate θε by φ

� � A � φ� 0 in � , φ� χ � ∂ηu0 on ∂ � .

� ai j � C1,β � χ j � C2,β � u1 � � k � H1 � 200 � � k �

� We decompose θε� ∑k � � e,w,n,s � θkε

Lεθkε

� 0 in � , θkε

�

��

�

� u1 � x, xε � � χ � ∂ηu0 on � k,

0 on ∂ � � � k .

approximate each θkε by φk

ε .

8


IMPA Versieux & Sarkis� Each function φk

ε is related to one side of the boundary and

decays exponentially to zero in the inner normal direction.

� Let Ge� � � � � , 0 � 0, 1 � and define ve

�� y

� a � y1 � 1, y2 � � yve � 0 in Ge,

ve � 0, � � � χ1 � 1, � � ,ve � � , 0 � � ve � � , 1 �

and ∂ve

∂yiexp � � γy1 � � L2 � Ge � .

Ge v e�χ

1

� An application of Tartar’s Lemma guaranties existence and

uniqueness of solution

9



� From Proposition 6.3 (Moscow and Vogelius (1997))� γ � 0, χ�e � � ve

� χ�e � exp � � γy1 � � L2 � Ge � ,

where

χ �e � 1

A11

1

0χ1 � 1, y2 � a1k � 1, y2 �

�

δk1

� ∂χ1 � 1, y2 �

∂y2 �

dy2

�

�� ,0 � � � 0,1 �� a � yve � � � yvedy

� We define,

φeε � x1, x2 � � ϕe � x1 � � ve � x1

� 1ε

, x2

ε � � χ �e ∂u0

∂x1 � x1, x2 � ,

ϕe is a cut off function, similar for φkε, k� � w, n, s � .

10

The Theoretical Approximation


� ϕe � 0 � � 0 andϕe � 1 � � 1 � φeε

� 0 on ∂ � � � e

� Define

φε� ∑k

φkε

� Approximate

uε� u0 � εu1 � ε � φε � φε �

Proposition 2 (Versieux/Sarkis (2005)). There exists a constant c

independent of u0 and ε, such that:

� uε � u0

� εu1

� ε � φε � φε � � 1 � cε � u0 � 2

� uε � u0

� εu1

� ε � φε � φε � � 0 � cε3 � 2

� u0 � 3

11

The Numerical Method


� χ j � A � u0 � vk � χ �k

� � � A � φ� 0 in � , φ� χ � ∂ηu0 on ∂ � � a good

approximation of ∂ηu0 is important.

� Find λhk � Vh

� � k � H1 � 200 � � k �

�

Ai j∂iuh0∂ jφdx�

�

fφdx �

� kλh

kφdσ , � φ� Vh, φ � ∂ � � � k

� 0

� Approximate ∂ηu0 by µh where

µh

� � k � λhk � All ,

��

�

l� 1 if k� e, w.

l� 2 if k� n, s.

12



� Approximate φ by a second order finite element

� Define

φe,hε � x1, x2 � � ϕe � x1 �

∂uh0

∂x1

� ve �

x1

� 1

ε,

x2

ε � � χ�e �

φhε

� ∑k

φk,hε

� Finally define

uhε

� uh0 � εuh

1 � ε � φhε � φh

ε �

� The same stiffness matrix is used to obtain u0 and φ

� The stiffness matrix for χ j is used to obtain vk

13

Convergence Analysis

IMPA Versieux & Sarkis� Theorem 1 (Versieux/Sarkis (2004)). Assume that ai j � C1,β � Y �

and that u0 � W2, � � � � , then there exists a constant c independent

of ε and h, such that� uε � uh � 1,h � c � h � ε � � u0 � 2, �

� Theorem 2 (Versieux/Sarkis (2004)). Assume that ai j � C1,β � Y �

and that u0 � W2, � � � � � H3 � � � , then there exists a constant C

independent of ε and h, such that

� uε � uh � 0 � C � h2

� ε32 � εh � � � u0 � 2, � � � u0 � 3 �

� O � h � ε � and O � h2

� ε32 � εhln � h � � if ∂ηuh

0 in place of µh

14

Numerical Results


Consider

a � x � �

2 � 1.8sin � 2πx1 � ε �

2 � 1.8cos � 2πx2 � ε ��

2 � sin � 2πx2 � ε �

2 � 1.8sin � 2πx1 � ε �

I2 � 2 , and f � � 1

the solution obtained by our method is compared with the solution

obtained by a second order accurate finite element method in a

mesh with size 1 � m f

�� 0 error, m f � 2048

ε � h � 1/8 1/16 1/32 1/64

1/16 2.7085e-04 7.7993e-05

1/32 2.6300e-04 6.6246e-05 1.7773e-05

1/64 2.5388e-04 5.9446e-05 1.4414e-05 1.2137e-05

15

Numerical Results


��

� 1,h error, m f � 2048

ε � h � 1/8 1/16 1/32 1/64

1/16 0.0097 0.0066

1/32 0.0089 0.0051 0.0036

1/64 0.0086 0.0045 0.0026 0.0018

� In order to obtain results with h � ε non-integer.

�� 0 and �� 1,h error, ε� 1 � 64, m f � 2240

h � 1/20 1/40

�� 0 3.6530e-05 1.2118e-05

�� 1,h 0.0037 0.0023

16

Numerical Results

IMPA Versieux & Sarkis� Error’s behavior as we include the terms from the asymptotic

expansion

ε� 1 � 64 h� 1 � 32, m f � 1024

�� 0 ��

� 1,h

uε � u0 0.0287 0.0215

uε � u0

� εu1 0.0213 0.0026

uε � u0

� εu1

� εφ 6.1557e-05 0.0026

uε � u0

� εu1

� ε � φ � φε � 6.1557e-05 0.0024

17

Conclusion and Ongoing work

� Conclusions

– The method does not require heavy computer effort. We

used a standard PC to run our method and PETSc on a 16

nodes cluster for the fine solution.

– The numerical results agree with the theoretical

convergence rate obtained in Theorem 1 and Theorem 2

� Ongoing work

– Include in the analysis the error due to the approximation

of χ j and vi.

– Stochastic case

18

Fine mesh approximation: uh fε , ε� 1 � 20, h f � 1 � 100


0

0.2

0.4

0.6

0.8

1

00.2

0.40.6

0.81

−0.02

−0.018

−0.016

−0.014

−0.012

−0.01

−0.008

−0.006

−0.004

−0.002

0

19

Error: uh fε

� uh0 , h� 10

� 1


00.2

0.40.6

0.81

0

0.2

0.4

0.6

0.8

1−1.5

−1

−0.5

0

0.5

1

1.5

2

x 10−3

20

Error: uh fε

� uh0

� εuh1, h� 10

� 1


00.2

0.40.6

0.81

00.2

0.40.6

0.81

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

x 10−3

21

Error: uh fε

� uh0

� εuh1

� εφh, h� 10

� 1


00.2

0.40.6

0.81

0

0.2

0.4

0.6

0.8

1−8

−6

−4

−2

0

2

4

6

x 10−4

22

Error: uh fε

� uh0

� εuh1

� ε � φh

� φh � , h� 10

� 1


00.2

0.40.6

0.81

0

0.2

0.4

0.6

0.8

1−8

−6

−4

−2

0

2

4

x 10−4

23

Bibliography


M. AVELLANEDA, F. H. LIN, Homogenization of Elliptic Problem with

Lp Boundary Data, Applied Mathematics and Optimization, 15

(1987), pp. 93-107.

A. BENSOUSSAN,J. L. LIONS AND G. PAPANICOLAOU, Asymptotic

Analysis for Periodic Structures, North Holland, Amsterdam (1980).

Y. R. EFENDIEV, T. HOU AND X. H. WU. Convergence of

Nonconforming Multi-scale Finite Element Method. SIAM J. Numer

Anal, 37 : 888-910, 2000.

T. HOU, X. H. WU, A Multi-scale Finite Element Method for Elliptic

Problems In Composite Materials and Porous Media , J. of Comp. Phys,

134 (1997), pp. 169-189.

24

Bibliography


T. HOU, X. H. WU, Z. CAI Convergence of Multi-scale Finite Element

Method for Elliptic Problems with Rapidly Oscillating Coefficients.

Mathematics of Computation, 68: 913-943, 1999.

S. MOSCOW, M. VOGELIUS , First-order corrections to the

homogenized eigenvalues of a periodic composite medium. A convergence

proof, Proceedings of the Royal Society of Edinburgh. 127A (1997)

1263-1299.

G. SANGALLI, Capturing Small Scales in Elliptic Problems Using a

Residual-Free Bubbles Finite Element Method, SIAM Multi-scale

Model. Simul., 1 (2003) pp 485-503

M. SARKIS AND H. M. VERSIEUX, Numerical Boundary Corrector for

Elliptic Equations with Rapidly Oscillating Coefficients, Submitted to

Communications in Numerical Methods in Engineering

25

Bibliography


M. SARKIS AND H. M. VERSIEUX, Convergence Analysis of a

Numerical Boundary Corrector for Elliptic Equations with Rapidly

Oscillating Coefficients, Paper in preparation

P. PEISKER. On the Numerical Solution of the First Biharmonic

Equation. Rairo Mathematical Modeling and Numerical Analysis

(1988); 22 655–676

J. L. LIONS , Some Methods for the Mathematical Analysis of Systems ,

Beejing: Science Press, and New York : Gordon and Breach, (1981).

O. A. LADYZHENKAYA, N. N. URAL’TSEVA. Equations aux derivees

Partielles de type Elliptique. Dunod, Paris (1968).

PETSc Web page.

http://www.mcs.anl.gov/petsc,

26



� Solve � y� a � y � � yχ

j� ∂xiai j � y � with a second order accurate

finite element in a mesh of size h, call these solutions by χj

h

�

Ahi j

� 1

�Y � Yaik � y � � δk j

� ∂∂yk

χj

h � dy.

� Let uh,h0 be the numerical approximation for

� � .Ah

� u0� f in � u0 � 0 on ∂ � ,

using a second order accurate finite element in a mesh � h � � � .

�

uh,h1 � x � � � χ j

h �

x

ε�

∂uh,h0

∂x j

� x � ,

27



� Let p� N, Gpe � � � p, 0 � 0, 1 and we� H1 � G

pe � , such that

�� y

� a � y1, y2 � � ywe � 0 in Gpe ,

we � 0, � � � χ1h � 1, � � ,

∂ηwe � 0 on � y1� � p � ,

and we � � , 0 � � we � � , 1 � .

Gpe w

e�χ

1h

∂ ηw

e�

0

� Call vh,pe the numerical approximate for the above problem

using a second order accurate finite element in a mesh of size h.

�

χ

� ,h,pe � 1

Ah11

1

0χ1

h � 1, y2 � a1k � 1, y2 ��

δk1

� ∂χ1h � 1, y2 �

∂y2

�

dy2

�

Gpe

� a � y1, y2 � � yvh,pe

� � yvh,pe � dy.

28


IMPA Versieux & Sarkis� � � A � φ� 0 in � , φ� χ � ∂ηu0 on ∂ � � a good

approximation of ∂ηu0 is important.

� Find λhk � Vh

� � k � H1 � 200 � � k �

�

Ai j∂iuh0∂ jφdx�

�

fφdx �

� kλh

kφdσ , � φ� Vh; φ � ∂ � � � k

� 0

� approximate ∂ηu0 by µh where

µh

� � k � λhk � All ,

��

�

l� 1 if k� e, w.

l� 2 if k� n, s.

29


IMPA Versieux & Sarkis� Let Ee � ∂x1

uh0 ,µh

e � � Vh be the extension of � ∂x1uh

0

� µh � � � e to

zero on � h � � � .

� Define

φe,h,h,pε � x1, x2 � �

��

��

�

ϕe � x1 � � vh,pe � x1

� 1ε

, x2

ε � � χ

� ,h,pe �

∂uh,h0

∂x1

� Ee � ∂x1uh,h

0 ,µhe � χ

� ,h,pe , if x1 � 1 � εp,

0, if x1 � 1 � εp,

� We approximate θε

φh,h,pε � ∑

k

φk,h,h,pε ,

30


IMPA Versieux & Sarkis� To approximate φ solve

� � Ah� ψ� 0 in � ψ� χ � ,h,pµh on ∂ � ,

with a second order finite element in a mesh of size h

� Finally define

uh,h,pε � uh,h

0 � εuh,h1 � ε � φ

h,h,pε � φ

h,h,pε �

� The same stiffness matrix is used to obtain u0 and φ.

� The stiffness matrix for χ j is used to obtain vk

31

homogenization finite elements · impa versieux & sarkis 1u0 h1 0 h2 ∂u0 ∂x n s 0 and ∂u0...

Documents