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