[Analyse a posteriori de la discrétisation d'un schéma multi-échelles pour des problèmes d'homogénéisation]
Dans cette Note, nous proposons une analyse a posteriori d'un schéma multi-échelles de type « micro–macro » pour des problèmes d'homogénéisation. Les paramètres du schéma macroscopique, inconnus à priori, sont obtenus pendant l'assemblage du problème homogénéisé à l'aide de schémas microscopiques. Le cadre que nous proposons pour l'analyse du schéma multi-échelles nous permet d'utiliser des techniques standards pour obtenir des indicateurs a posteriori par résidu de l'erreur. Ces indicateurs d'erreur permettent de mettre en oeuvre une stratégie d'adaptation du maillage.
In this Note we derive a posteriori error estimates for a multiscale method, the so-called heterogeneous multiscale method, applied to elliptic homogenization problems. The multiscale method is based on a macro-to-micro formulation. The macroscopic method discretizes the physical problem in a macroscopic finite element space, while the microscopic method recovers the unknown macroscopic data on the fly during the macroscopic stiffness matrix assembly process. We propose a framework for the analysis allowing to take advantage of standard techniques for a posteriori error estimates at the macroscopic level and to derive residual-based indicators in the macroscopic domain for adaptive mesh refinement.
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@article{CRMATH_2009__347_17-18_1081_0,
author = {Assyr Abdulle and Achim Nonnenmacher},
title = {A posteriori error analysis of the heterogeneous multiscale method for homogenization problems},
journal = {Comptes Rendus. Math\'ematique},
pages = {1081--1086},
publisher = {Elsevier},
volume = {347},
number = {17-18},
year = {2009},
doi = {10.1016/j.crma.2009.07.004},
language = {en},
}
TY - JOUR AU - Assyr Abdulle AU - Achim Nonnenmacher TI - A posteriori error analysis of the heterogeneous multiscale method for homogenization problems JO - Comptes Rendus. Mathématique PY - 2009 SP - 1081 EP - 1086 VL - 347 IS - 17-18 PB - Elsevier DO - 10.1016/j.crma.2009.07.004 LA - en ID - CRMATH_2009__347_17-18_1081_0 ER -
%0 Journal Article %A Assyr Abdulle %A Achim Nonnenmacher %T A posteriori error analysis of the heterogeneous multiscale method for homogenization problems %J Comptes Rendus. Mathématique %D 2009 %P 1081-1086 %V 347 %N 17-18 %I Elsevier %R 10.1016/j.crma.2009.07.004 %G en %F CRMATH_2009__347_17-18_1081_0
Assyr Abdulle; Achim Nonnenmacher. A posteriori error analysis of the heterogeneous multiscale method for homogenization problems. Comptes Rendus. Mathématique, Volume 347 (2009) no. 17-18, pp. 1081-1086. doi : 10.1016/j.crma.2009.07.004. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2009.07.004/
[1] On a-priori error analysis of fully discrete heterogeneous multiscale FEM, SIAM Multiscale Model. Simul., Volume 4 (2005) no. 2, pp. 447-459
[2] Analysis of a heterogeneous multiscale FEM for problems in elasticity, Math. Mod. Meth. Appl. Sci. (M3AS), Volume 16 (2006) no. 4, pp. 615-635
[3] Multiscale method based on discontinuous Galerkin methods for homogenization problems, C. R. Acad. Sci. Paris, Ser. I, Volume 346 (2008), pp. 9-102
[4] The finite element heterogeneous multiscale method: A computational strategy for multiscale PDEs, GAKUTO Int. Ser. Math. Sci. Appl., Volume 31 (2009), pp. 135-184
[5] Finite element heterogeneous multiscale methods with near optimal computational complexity, SIAM Multiscale Model. Simul., Volume 6 (2007) no. 4, pp. 1059-1084
[6] A. Abdulle, A. Nonnenmacher, A short and versatile finite element multiscale code for homogenization problems, Comput. Methods Appl. Mech. Engrg., | DOI
[7] A. Abdulle, A. Nonnenmacher, Adaptive finite element heterogeneous multiscale method for homogenization problems, in preparation
[8] Heterogeneous multiscale FEM for diffusion problem on rough surfaces, SIAM Multiscale Model. Simul., Volume 3 (2005) no. 1, pp. 195-220
[9] L. Chenand, C.S. Zhang, AFEM@Matlab: A Matlab package of adaptive finite element methods, Technical report, 2006
[10] A convergent adaptive algorithm for Poisson's equation, SIAM J. Numer. Anal., Volume 33 (1996), pp. 1106-1124
[11] The heterogeneous multi-scale method, Commun. Math. Sci., Volume 1 (2003) no. 1, pp. 87-132
[12] 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
[13] Analysis of the heterogeneous multiscale method for elliptic homogenization problems, J. Amer. Math. Soc., Volume 18 (2004) no. 1, pp. 121-156
[14] Homogenization of Differential Operators and Integral Functionals, Springer-Verlag, 1994
[15] A framework for adaptive multiscale methods for elliptic problems, SIAM Multiscale Model. Simul., Volume 7 (2008), pp. 171-196
[16] A Posteriori Error Estimation in Finite Element Analysis, John Wiley & Sons, New York, 2000
[17] A posteriori error estimates for the heterogeneous multiscale finite element method for elliptic homogenization problems, SIAM Multiscale Model. Simul., Volume 4 (2005) no. 2, pp. 88-114
[18] A Review of a Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques, Wiley–Teubner, New York, 1996
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