The Multi-Grid Method for the Diffusion Equation with Strongly Discontinuous Coefficients

Abstract

The subject of this paper is the application of the multi-grid method to the solution of \[ - \nabla \cdot (D(x,y)\nabla U(x,y)) + \sigma (x,y)U(x,y) = f(x,y) \] in a bounded region $\Omega $ of $R^2 $ where D is positive and D, $\sigma $, and f are allowed to be discontinuous across internal boundaries $\Gamma $ of $\Omega $. The emphasis is on discontinuities of orders of magnitude in D, when special techniques must be applied to restore the multi-grid method to good efficiency. These techniques are based on the continuity of $D\nabla U$ across $\Gamma $. Two basic methods are derived, one in which the approximating finite difference operators on coarser grids are five point operators (assuming the finite difference operator on the finest grid is a five point one) and one in which they are nine point operators.