An Isaac Newton Institute Workshop

The Theory of Highly Oscillatory Problems

Finite difference approximation of homogenization problems for elliptic equations

Author: Enrique Zuazua (Universidad Autónoma)

Abstract

In this talk, the problem of the approximation by finite

differences of solutions to elliptic problems with rapidly

oscillating coefficients and periodic boundary conditions will be discussed.

The mesh-size is denoted by $h$ while $\ee$

denotes the period of the rapidly oscillating coefficient. Using

Bloch wave decompositions, we analyze the case where the ratio

$h/\ee$ is rational. We show that if $h/\ee$ is kept fixed, being

a rational number, even when $h,\ee\to 0$, the limit of the

numerical solution does not coincide with the homogenized one

obtained when passing to the limit as $\ee\to 0$ in the continuous

problem. Explicit error estimates are given showing that, as the

ratio $h/\ee$ approximates an irrational number, solutions of the

finite difference approximation converge to the solutions of the

homogenized elliptic equation. We consider both the 1-d and the

multi-dimensional case. Our analysis yields a quantitative version

of previous results on numerical homogenization by Avellaneda, Hou

and Papanicolaou, among others. This is a joint work with Rafael Orive