A Finite Difference Scheme for Elliptic Equations with Rough Coefficients Using a Cartesian Grid Nonconforming to Interfaces

Abstract

We consider the problem of calculating a potential function in a two-dimensional inhomogeneous medium which varies locally in only one direction. We propose a staggered finite difference scheme on a regular Cartesian grid with a special cell averaging. This averaging allows for the change in conductivity to be in any direction with respect to the grid and does not require the grid to be small compared to the layering. We give a convergence result and numerical experiments which suggest that the new averaging works as well as the standard homogenization with thin conductive nonconformal sheets and exhibits better accuracy for resistive sheets.