# HOP

## Seminar

### Finite difference approximation of homogenization problems for elliptic equations

Seminar Room 1, Newton Institute

#### 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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