Optimal Approximation of the First-Order Corrector in Multiscale Stochastic Elliptic PDE

Abstract

This work addresses the development of an optimal computational scheme for the approximation of the first-order corrector arising in the stochastic homogenization of linear elliptic PDEs in diver- gence form. Equations of this type describe, for example, diffusion phenomena in materials with a heterogeneous microstructure, but require enormous computational efforts in order to obtain reliable results. We derive an optimization problem for the needed computational work with a given error tolerance, then extract the governing parameters from numerical experiments, and finally solve the obtained optimization problem. The numerical approach investigated here is a stochastic sampling scheme for the probability space connected with a finite-element method for the discretization of the physical space.