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On a variant of random homogenization theory: convergence of the residual process and approximation of the homogenized coefficients

Frédéric Legoll 1, 2 Florian Thomines 1, 3
1 MATHERIALS - MATHematics for MatERIALS
ENPC - École des Ponts ParisTech, CERMICS - Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique, Inria Paris-Rocquencourt
Abstract : We consider the variant of stochastic homogenization theory introduced in [X. Blanc, C. Le Bris and P.-L. Lions, C. R. Acad. Sci. Serie I 2006 and Journal de Mathematiques Pures et Appliquees 2007]. The equation under consideration is a standard linear elliptic equation in divergence form, where the highly oscillatory coefficient is the composition of a periodic matrix with a stochastic diffeomorphism. The homogenized limit of this problem has been identified in [X. Blanc, C. Le Bris and P.-L. Lions, C. R. Acad. Sci. Serie I 2006]. We first establish, in the one-dimensional case, a convergence result (with an explicit rate) on the residual process, defined as the difference between the solution to the highly oscillatory problem and the solution to the homogenized problem. We next return to the multidimensional situation. As often in random homogenization, the homogenized matrix is defined from a so-called corrector function, which is the solution to a problem set on the entire space. We describe and prove the almost sure convergence of an approximation strategy based on truncated versions of the corrector problem.
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https://hal-enpc.archives-ouvertes.fr/hal-00755230
Contributeur : Frederic Legoll <>
Soumis le : mardi 20 novembre 2012 - 17:00:13
Dernière modification le : vendredi 17 juillet 2020 - 17:09:09

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  • HAL Id : hal-00755230, version 1
  • ARXIV : 1211.4252

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Frédéric Legoll, Florian Thomines. On a variant of random homogenization theory: convergence of the residual process and approximation of the homogenized coefficients. ESAIM: Mathematical Modelling and Numerical Analysis, EDP Sciences, 2014, 48 (2), pp.347-386. ⟨hal-00755230⟩

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