Fourth‐order finite difference method for three‐dimensional elliptic equations with nonlinear first‐derivative terms

Abstract

We present a 19‐point fourth‐order finite difference method for the nonlinear second‐order system of three‐dimensional elliptic equations Auxx + Buyy + Cuzz = f, where A, B, C, are M × M diagonal matrices, on a cubic region R subject to the Dirichlet boundary conditions u(x, y, z) = u0(x, y, z) on ∂R. We establish, under appropriate conditions, O(h4) convergence of the difference method. Numerical examples are given to illustrate the method and its fourth‐order convergence. © 1992 John Wiley & Sons, Inc. Copyright © 1992 Wiley Periodicals, Inc.