An Embedded Corrector Problem for Homogenization. Part I: Theory

Multiscale Modeling and Simulation, vol. 18, pp. 1179–1209

Abstract

This article is the first part of a two-fold study, the objective of which is the theoretical analysis and numerical investigation of new approximate corrector problems in the context of stochastic homogenization. We present here three new alternatives for the approximation of the homogenized matrix for diffusion problems with highly oscillatory coefficients. These different approximations all rely on the use of an embedded corrector problem (that we previously introduced in [Cancès et al., C. R. Math. Acad. Sci. Paris, 353 (2015), pp. 801--806]), where a finite-size domain made of the highly oscillatory material is embedded in a homogeneous infinite medium whose diffusion coefficients have to be appropriately determined. The motivation for considering such embedded corrector problems is made clear in the companion article [Cancès et al., J. Comput. Phys., 407 (2020), 109254], where a very efficient algorithm is presented for the resolution of such problems for particular heterogeneous materials. In the present article, we prove that the three different approximations we introduce converge to the homogenized matrix of the medium when the size of the embedded domain goes to infinity.

Authors 5

1. Éric Cancès

   CERMICS · MATHERIALS: MATHematics for MatERIALS

   Affiliation as printed

   CERMICS - Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique (6 et 8 avenue Blaise Pascal Cité Descartes - Champs sur Marne 77455 Marne la Vallée Cedex 2 - France)

   MATHERIALS - MATHematics for MatERIALS (France)

2. Virginie Ehrlacher

   CERMICS · MATHERIALS: MATHematics for MatERIALS

   Affiliation as printed

   CERMICS - Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique (6 et 8 avenue Blaise Pascal Cité Descartes - Champs sur Marne 77455 Marne la Vallée Cedex 2 - France)

   MATHERIALS - MATHematics for MatERIALS (France)

3. Frédéric Legoll

   École nationale des ponts et chaussées · Laboratoire Navier · MATHERIALS: MATHematics for MatERIALS

   Affiliation as printed

   MATHERIALS - MATHematics for MatERIALS (France)

   NAVIER UMR 8205 - Laboratoire Navier (École des Ponts ParisTech, 6-8 avenue Blaise Pascal, 77455 Champs-sur-Marne - France)

4. Benjamin Stamm Aachen

   RWTH Aachen University

   Affiliation as printed

   RWTH - Rheinisch-Westfälische Technische Hochschule Aachen University (RWTH Aachen Templergraben 55 52062 Aachen (Hausanschrift) 52056 Aachen (Postanschrift) - Germany)

5. Shuyang Xiang

   CERMICS

   Affiliation as printed

   CERMICS - Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique (6 et 8 avenue Blaise Pascal Cité Descartes - Champs sur Marne 77455 Marne la Vallée Cedex 2 - France)