Comptes Rendus
no. 6
Numerical analysis
Exponential decay of the resonance error in numerical homogenization via parabolic and elliptic cell problems
[Décroissance exponentielle de l'erreur de résonance en homogénéisation numerique via des problèmes de cellules paraboliques et elliptiques]

Présenté par : the Editorial Board

Assyr Abdulle1 ; Doghonay Arjmand1 ; Edoardo Paganoni1
1 ANMC, Institut de mathématiques, École polytechnique fédérale de Lausanne, CH-1015 Lausanne, Switzerland
Comptes Rendus. Mathématique, Volume 357 (2019) no. 6, pp. 545-551.

Cette note présente deux nouvelles approches pour trouver les coefficients homogénéisés des EDP elliptiques multi-échelles. Les approches standard pour calculer les coefficients homogénéisés souffrent de ce que l'on appelle l'erreur de résonance, qui découle d'une inadéquation entre les vraies conditions aux limites et celles computationelles. Nos nouvelles méthodes, basées sur des solutions aux problèmes de cellules paraboliques et elliptiques, entraînent une décroissance exponentielle de l'erreur de résonance.

This paper presents two new approaches for finding the homogenized coefficients of multiscale elliptic PDEs. Standard approaches for computing the homogenized coefficients suffer from the so-called resonance error, originating from a mismatch between the true and the computational boundary conditions. Our new methods, based on solutions of parabolic and elliptic cell problems, result in an exponential decay of the resonance error.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2019.05.011
Assyr Abdulle 1 ; Doghonay Arjmand 1 ; Edoardo Paganoni 1

1 ANMC, Institut de mathématiques, École polytechnique fédérale de Lausanne, CH-1015 Lausanne, Switzerland 
@article{CRMATH_2019__357_6_545_0,
     author = {Assyr Abdulle and Doghonay Arjmand and Edoardo Paganoni},
     title = {Exponential decay of the resonance error in numerical homogenization via parabolic and elliptic cell problems},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {545--551},
     publisher = {Elsevier},
     volume = {357},
     number = {6},
     year = {2019},
     doi = {10.1016/j.crma.2019.05.011},
     language = {en},
}
TY  - JOUR
AU  - Assyr Abdulle
AU  - Doghonay Arjmand
AU  - Edoardo Paganoni
TI  - Exponential decay of the resonance error in numerical homogenization via parabolic and elliptic cell problems
JO  - Comptes Rendus. Mathématique
PY  - 2019
SP  - 545
EP  - 551
VL  - 357
IS  - 6
PB  - Elsevier
DO  - 10.1016/j.crma.2019.05.011
LA  - en
ID  - CRMATH_2019__357_6_545_0
ER  - 
%0 Journal Article
%A Assyr Abdulle
%A Doghonay Arjmand
%A Edoardo Paganoni
%T Exponential decay of the resonance error in numerical homogenization via parabolic and elliptic cell problems
%J Comptes Rendus. Mathématique
%D 2019
%P 545-551
%V 357
%N 6
%I Elsevier
%R 10.1016/j.crma.2019.05.011
%G en
%F CRMATH_2019__357_6_545_0
Assyr Abdulle; Doghonay Arjmand; Edoardo Paganoni. Exponential decay of the resonance error in numerical homogenization via parabolic and elliptic cell problems. Comptes Rendus. Mathématique, Volume 357 (2019) no. 6, pp. 545-551. doi : 10.1016/j.crma.2019.05.011. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2019.05.011/

[1] A. Abdulle Fourth order Chebyshev methods with recurrence relation, SIAM J. Sci. Comput., Volume 23 (2002) no. 6, pp. 2041-2054 | DOI

[2] A. Abdulle, D. Arjmand, E. Paganoni, Reduction of the modelling error in numerical homogenization problems: a parabolic approach, preprint.

[3] A. Abdulle, D. Arjmand, E. Paganoni, A fully elliptic local problem with exponential decay for numerical homogenization problems, preprint.

[4] A. Abdulle; W. E; B. Engquist; E. Vanden-Eijnden The heterogeneous multiscale method, Acta Numer., Volume 21 (2012), pp. 1-87 | DOI

[5] D. Arjmand; O. Runborg A time dependent approach for removing the cell boundary error in elliptic homogenization problems, J. Comput. Phys., Volume 341 (2016), pp. 206-227 | DOI

[6] A. Bensoussan; J.-L. Lions; G. Papanicolaou Asymptotic Analysis for Periodic Structures, North-Holland, Amsterdam, 1978

[7] X. Blanc; C. Le Bris Improving on computation of homogenized coefficients in the periodic and quasi-periodic settings, Netw. Heterog. Media, Volume 5 (2010) no. 1, pp. 1-29 | DOI

[8] A. Bourgeat; A. Piatniski Approximation of effective coefficients in stochastic homogenization, Ann. Inst. Henri Poincaré Probab. Stat., Volume 40 (2004) no. 2, pp. 153-165 | DOI

[9] D. Cioranescu; P. Donato An Introduction to Homogenization, Oxford University Press, 1999

[10] W. E; P. Ming; P. Zhang Analysis of the heterogeneous multiscale method for elliptic homogenization problems, J. Amer. Math. Soc., Volume 18 (2005), pp. 121-156 | DOI

[11] A. Gloria Reduction of the resonance error. Part 1: approximation of homogenized coefficients, Math. Models Methods Appl. Sci., Volume 21 (2011) no. 8, pp. 1601-1630 | DOI

[12] A. Gloria; F. Otto An optimal error estimate in stochastic homogenization of discrete elliptic equations, Ann. Appl. Probab., Volume 22 (2012) no. 1, pp. 1-28 | DOI

[13] V. Jikov; S.M. Kozlov; O.A. Oleinik Homogenization of Differential Operators and Integral Functionals, Springer-Verlag, 1994

[14] J.-C. Mourrat Efficient methods for the estimation of homogenized coefficients, Found. Comput. Math., Volume 19 (2019) no. 2, pp. 435-483 | DOI

[15] X. Yue; W. E The local microscale problem in the multiscale modeling of strongly heterogeneous media: effects of boundary conditions and cell size, J. Comput. Phys., Volume 222 (2007) no. 2, pp. 556-572 | DOI

[16] J.G. Verwer; W.H. Hundsdorfer; B.P. Sommeijer Convergence properties of the Runge-Kutta-Chebyshev method, Numer. Math., Volume 57 (1990) no. 1, pp. 157-178 | DOI

Cité par Sources :

☆ This work is partially supported by the Swiss National Science Foundation, grant No. 200020_172710. The authors thank Jean-Christophe Mourrat for useful discussions.

Commentaires - Politique