Asymptotic expansion of the homogenized matrix in two weakly stochastic homogenization settings

TL;DR

This paper develops an asymptotic expansion for the approximated homogenized matrix in stochastic elliptic PDEs with weak randomness, accounting for discretization effects and randomness in the expansion coefficients.

Contribution

It extends theoretical expansions of the homogenized matrix to practical numerical approximations, including the randomness of expansion coefficients.

Findings

Derived an expansion for the approximated homogenized matrix including random coefficients.

Analyzed how discretization parameters affect the expansion.

Provided insights into the second order term's stochastic nature.

Abstract

This article studies some numerical approximations of the homogenized matrix for stochastic linear elliptic partial differential equations in divergence form. We focus on the case when the underlying random field is a small perturbation of a reference periodic tensor. The size of such a perturbation is encoded by a real parameter eta. In this case, it has already been theoretically shown in the literature that the exact homogenized matrix possesses an expansion in powers of the parameter eta for both models considered in this article, the coefficients of which are deterministic. In practice, one cannot manipulate the exact terms of such an expansion. All objects are subjected to a discretization approach. Thus we need to derive a similar expansion for the approximated random homogenized matrix. In contrast to the expansion of the exact homogenized matrix, the expansion of the approximated homogenized matrix contains intrinsically random coefficients. In particular, the second order term is random in nature. The purpose of this work is to derive and study this expansion in function of the parameters of the approximation procedure (size of the truncated computational domain used, meshsize of the finite elements approximation).