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Approximations of effective coefficients in stochastic homogenization. (English) Zbl 1058.35023
The main goal of this paper is to provide a rigorous mathematical justification for the convergence and to give, whenever possible, estimates for the rate of convergence of the various localization methods used in engineering literature to approximate the effective tensor of random stationary media. Under the additional uniform mixing conditions on coefficients, the authors estimate the rates of convergence of approximations and show that the corresponding bounds only depend on the ellipticity constant in the original problem, the space dimension, and the rate of decay of the uniform mixing coefficients. To this end, the authors penalize the original operator by adding a small positive potential and introduce “effective auxiliary characteristics” associated to this penalized operator.

MSC:
35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
35K20 Initial-boundary value problems for second-order parabolic equations
35Q35 PDEs in connection with fluid mechanics
35R60 PDEs with randomness, stochastic partial differential equations
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