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.
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.
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@article{CRMATH_2009__347_17-18_1081_0,
author = {Abdulle, Assyr and Nonnenmacher, Achim},
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},
url = {http://www.numdam.org/articles/10.1016/j.crma.2009.07.004/}
}
TY - JOUR AU - Abdulle, Assyr AU - Nonnenmacher, Achim 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 UR - http://www.numdam.org/articles/10.1016/j.crma.2009.07.004/ DO - 10.1016/j.crma.2009.07.004 LA - en ID - CRMATH_2009__347_17-18_1081_0 ER -
%0 Journal Article %A Abdulle, Assyr %A Nonnenmacher, Achim %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 %U http://www.numdam.org/articles/10.1016/j.crma.2009.07.004/ %R 10.1016/j.crma.2009.07.004 %G en %F CRMATH_2009__347_17-18_1081_0
Abdulle, Assyr; Nonnenmacher, Achim. 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. http://www.numdam.org/articles/10.1016/j.crma.2009.07.004/
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