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

The subject of this paper is the application of the multi-grid method to the solution of \[ - ∇ · (D(x,y)∇ U(x,y)) + σ (x,y)U(x,y) = f(x,y) \] in a bounded region Ω of R² where D is positive and D, σ, and f are allowed to be discontinuous across internal boundaries Γ of Ω. 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∇ U across Γ. 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.