# 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

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