Asymptotically Compatible Schemes for Stochastic Homogenization
Qi Yu Sun, Qiang Du, Ming Ju · SIAM Journal on Numerical Analysis · 2018
In this paper, we study the numerical solution of elliptic homogenization problems with stationary and ergodic coefficients characterized by a small length scale $\varepsilon$. The issue of asymptotic compatibility as $\varepsilon \to 0$ is examined for a number of different numerical approximations, which enables us to construct robust discretization schemes to eliminate the resonance and under-resolution errors caused by the inappropriate use of base functions. In addition to the study of effective characteristics based on particular sample realizations that are subject to random effects, we also consider approximations by numerically evaluating the expectation of realized values. When the Monte Carlo finite element method is applied for the latter, error estimation is derived to guide parameter choices so as to further enhance the computational efficiency by performing relatively few samples while attaining high accuracy. Numerical experiments are presented to validate and supplement our theoretical findings.