Finite Difference Approximation of Homogenization Problems for Elliptic Equations

In this paper, the problem of the approximation by finite differences of solutions to elliptic problems with rapidly oscillating coefficients and periodic boundary conditions is considered. The mesh-size is denoted by h while ε denotes the period of the rapidly oscillating coefficient. Using Bloch wave decompositions, we analyze the case where the ratio h/ε is rational. We show that if h/ε is kept fixed, being a rational number, even when h, ε → 0, the limit of the numerical solution does not coincide with the homogenized one obtained when passing to the limit as ε → 0 in the continuous problem. Explicit error estimates are given showing that, as the ratio h/ε 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 M. Avellaneda, Th.Y. Hou and G. Papanicolaou [1].