Asymptotic Expansion of the Homogenized Matrix in Two Weakly Stochastic Homogenization Settings
Ronan Costaouec · Applied Mathematics Research eXpress · 2011
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 A is a small perturbation Aη of a reference periodic tensor Aper, where η encodes the size of the perturbation. In this case, it has been theoretically shown in Blanc et al. [5, 7] for both models considered in this article that the exact homogenized matrix possesses an expansion in powers of η, 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 for the variables x (Finite Element method) and ω (Monte-Carlo method). 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 approximate 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, mesh size of the finite elements approximation).