Partial differential equations/Numerical analysis
An embedded corrector problem to approximate the homogenized coefficients of an elliptic equation
Comptes Rendus. Mathématique, Volume 353 (2015) no. 9, pp. 801-806.

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 R. We also discuss efficient numerical methods to solve the embedded corrector problem.

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 R vers le coefficient homogénéisé. Nous mentionnons également des méthodes de résolution numérique efficaces pour ce nouveau problème.

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Accepted:
Published online:
DOI: 10.1016/j.crma.2015.06.019
Cancès, Éric 1, 2; Ehrlacher, Virginie 1, 2; Legoll, Frédéric 3, 2; Stamm, Benjamin 4

1 CERMICS, École des Ponts ParisTech, 77455 Marne-la-Vallée cedex 2, France
2 INRIA Rocquencourt, MATHERIALS project-team, Domaine de Voluceau, BP 105, 78153 Le Chesnay cedex, France
3 Laboratoire Navier, École des Ponts ParisTech, 77455 Marne-la-Vallée cedex 2, France
4 Sorbonne Universités, UPMC Université Paris-6 and CNRS, UMR 7598, Laboratoire Jacques-Louis Lions, 75005 Paris, France
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Cancès, Éric; Ehrlacher, Virginie; Legoll, Frédéric; Stamm, Benjamin. An embedded corrector problem to approximate the homogenized coefficients of an elliptic equation. Comptes Rendus. Mathématique, Volume 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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Cited by Sources:

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.