Error estimates and optimal approaches to the stochastic homogenization in elliptic partial differential equations

Abstract

This thesis is focused on the numerical treatment of stochastic homogenization in elliptic partial differential equations. A major application, and one that we keep in mind in this work, can be found in materials science, where one is interested in obtaining an average conductivity for a composite containing a fine microscopic structure. Such problems are also called "multiscale" due to the differences in (length) scales that are of interest. Determining an average conductivity requires first resolving at least part of the fine structure, which we can use to estimate the conductivity on the macroscopic scale. A central issue in numerically solving a homogenization problem comes from the fact that in order to minimize the error, one would need to determine conductivity on the microscale for the entire body; this is however not possible given the sheer size of such systems. One needs to therefore content oneself with a sample of the material and use this information to compute an average conductivity. This can be accomplished by first solving the so-called "cell problem." The essential contribution in this thesis is the estimation of the error of the cell problem, which we express as a function of domain size, mesh fineness and number of samples. We will quantify the work needed to solve this problem and then present an optimal approach to solving the problem.