Journal
ANNALES DE L INSTITUT HENRI POINCARE-PROBABILITES ET STATISTIQUES
Volume 40, Issue 2, Pages 153-165Publisher
INST MATHEMATICAL STATISTICS
DOI: 10.1016/j.anihpb.2003.07.003
Keywords
random operator; volume averaging; homogenization
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This note deals with localized approximations of homogenized coefficients of second order divergence form elliptic operators with random statistically homogeneous coefficients, by means of periodization and other cut-off procedures. For instance in the case of periodic approximation, we consider a cubic sample [0, rho](d) of the random medium, extend it periodically in R-d and use the effective coefficients of the obtained periodic operators as an approximation of the effective coefficients of the original random operator. It is shown that this approximation converges a.s., as rho --> infinity, and gives back the effective coefficients of the original random operator. Moreover, under additional mixing conditions on the coefficients, the rate of convergence can be estimated by some negative power of rho which only depends on the dimension, the ellipticity constant and the rate of decay of the mixing coefficients. Similar results are established for approximations in terms of appropriate Dirichlet and Neumann problems localized in a cubic sample [0, rho](d). (C) 2004 Elsevier SAS. All rights reserved.
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