[Un problème d'inclusion pour approcher les coefficients homogénéisés d'une équation elliptique]
Nous considérons une équation de diffusion à coefficients hautement oscillants qui admet une limite homogénéisée, et nous introduisons une variante du problème du correcteur standard, qui se formalise comme un problème d'inclusion. Celui-ci s'écrit comme une équation de diffusion posée dans tout l'espace, dans laquelle la matrice de diffusion est uniforme à l'extérieur d'une boule de rayon R. Nous introduisons ensuite trois approximations des coefficients homogénéisés, calculées à partir de la solution de ce problème. Ces approximations, qui sont des variantes des approximations standard basées sur le problème du correcteur tronqué (méthode de supercellule), convergent lorsque vers le coefficient homogénéisé. Nous mentionnons également des méthodes de résolution numérique efficaces pour ce nouveau problème.
We consider a diffusion equation with highly oscillatory coefficients that admits a homogenized limit. As an alternative to standard corrector problems, we introduce here an embedded corrector problem, written as a diffusion equation in the whole space, in which the diffusion matrix is uniform outside some ball of radius R. Using that problem, we next introduce three approximations of the homogenized coefficients. These approximations, which are variants of the standard approximations obtained using truncated (supercell) corrector problems, are shown to converge to the homogenized coefficient when . We also discuss efficient numerical methods to solve the embedded corrector problem.
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@article{CRMATH_2015__353_9_801_0,
author = {Canc\`es, \'Eric and Ehrlacher, Virginie and Legoll, Fr\'ed\'eric and Stamm, Benjamin},
title = {An embedded corrector problem to approximate the homogenized coefficients of an elliptic equation},
journal = {Comptes Rendus. Math\'ematique},
pages = {801--806},
publisher = {Elsevier},
volume = {353},
number = {9},
year = {2015},
doi = {10.1016/j.crma.2015.06.019},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.crma.2015.06.019/}
}
TY - JOUR AU - Cancès, Éric AU - Ehrlacher, Virginie AU - Legoll, Frédéric AU - Stamm, Benjamin TI - An embedded corrector problem to approximate the homogenized coefficients of an elliptic equation JO - Comptes Rendus. Mathématique PY - 2015 SP - 801 EP - 806 VL - 353 IS - 9 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2015.06.019/ DO - 10.1016/j.crma.2015.06.019 LA - en ID - CRMATH_2015__353_9_801_0 ER -
%0 Journal Article %A Cancès, Éric %A Ehrlacher, Virginie %A Legoll, Frédéric %A Stamm, Benjamin %T An embedded corrector problem to approximate the homogenized coefficients of an elliptic equation %J Comptes Rendus. Mathématique %D 2015 %P 801-806 %V 353 %N 9 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2015.06.019/ %R 10.1016/j.crma.2015.06.019 %G en %F CRMATH_2015__353_9_801_0
Cancès, Éric; Ehrlacher, Virginie; Legoll, Frédéric; Stamm, Benjamin. An embedded corrector problem to approximate the homogenized coefficients of an elliptic equation. Comptes Rendus. Mathématique, Tome 353 (2015) no. 9, pp. 801-806. doi : 10.1016/j.crma.2015.06.019. http://www.numdam.org/articles/10.1016/j.crma.2015.06.019/
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☆ We thank Paul Cazeaux for fruitful discussions on questions related to this project. The work of FL is partially supported by ONR under Grant N00014-12-1-0383 and EOARD under Grant FA8655-13-1-3061.
