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Numerical approximation of effective coefficientsin stochastic homogenization of discrete elliptic equations

Published online by Cambridge University Press:  22 July 2011

Antoine Gloria
Antoine Gloria*
Affiliation:
Project-Team SIMPAF, INRIA Lille-Nord Europe, France and Laboratoire Paul Painlevé (UMR CNRS 8524), Université Lille 1, 59655 Villeneuve d'Ascq, France. [email protected]
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Abstract

We introduce and analyze a numerical strategy to approximate effective coefficients in stochastic homogenization of discrete ellipticequations. In particular, we consider the simplest case possible: An elliptic equation on the d-dimensional lattice $\mathbb{Z}^d$with independent and identically distributed conductivities on the associated edges.Recent results by Otto and the author quantify the error made by approximatingthe homogenized coefficient by the averaged energy of a regularizedcorrector (with parameter T) on some box of finite size L. In this article, we replace the regularizedcorrector (which is the solution of a problem posed on $\mathbb{Z}^d$) by some practically computable proxy on some box of size R ≥ L, and quantify the associated additional error.In order to improve the convergence, one may also consider N independentrealizations of the computable proxy, and take the empirical average of the associated approximate homogenized coefficients.A natural optimization problem consists in properly choosing T, R, L and N in order toreduce the error at given computational complexity.Our analysis is sharp and sheds some light on this question.In particular, we propose and analyze a numerical algorithm to approximate the homogenized coefficients, taking advantage of the (nearly) optimal scalings of the errorswe derive. The efficiency of the approach is illustrated by a numerical study in dimension 2.

Keywords

Stochastic homogenizationeffective coefficientsdifference operatornumerical method
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 46 , Issue 1 , January 2012 , pp. 1 - 38
Copyright
© EDP Sciences, SMAI, 2011

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