Preprint 28/2010

An optimal error estimate in stochastic homogenization of discrete elliptic equations

Antoine Gloria and Felix Otto

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Submission date: 26. May. 2010
Pages: 26
published in: The annals of applied probability, 22 (2012) 1, p. 1-28 
DOI number (of the published article): 10.1214/10-AAP745
MSC-Numbers: 35B27, 39A70, 60H25, 60F99
Keywords and phrases: stochastic homogenization, effective coefficients, difference operator
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Abstract:
This paper is the second of a series of articles on quantitatives estimates in stochastic homogenization of discrete elliptic equations. We consider a discrete elliptic equation on the d-dimensional lattice formula15 with random coefficients A of the simplest type: They are identically distributed and independent from edge to edge. On scales large w. r. t. the lattice spacing (i. e. unity), the solution operator is known to behave like the solution operator of a (continuous) elliptic equation with constant deterministic coefficients. This symmetric ``homogenized'' matrix formula19 is characterized by formula21 for any direction formula23, where the random field formula25 (the ``corrector'') is the unique solution of formula27 in formula29 such that formula31, formula33 is stationary and formula35, formula37 denoting the ensemble average (or expectation).

In order to approximate the homogenized coefficients formula39, the corrector problem is usually solved in a box formula41 of size 2L with periodic boundary conditions, and the space averaged energy on formula45 defines an approximation formula47 of formula39. Although the statistics is modified (independence is replaced by periodic correlations) and the ensemble average is replaced by a space average, the approximation formula47 converges almost surely to formula39 as formula55. In this paper, we give estimates on both errors. To be more precise, we do not consider periodic boundary conditions on a box of size 2L, but replace the elliptic operator by formula59 with (typically) formula61, as standard in the homogenization literature. We then replace the ensemble average by a space average on formula45, and estimate the overall error on the homogenized coefficients in terms of L and T.

19.04.2013, 01:42