Some convergence results in elliptic difference equations

Abstract

The purpose of the paper is to present a survey of some recent results in convergence theory for finite-difference approximations to Dirichlet’s problem for second-order linear elliptic differential equations. The basic approach is to first derive a priori inequalities for the discrete problem and then to use these to deduce corresponding convergence estimates. First difference approximations of positive type and related maximum-norm estimates are considered. Then some schemes of non-positive type are described and analysed by L 2 methods. Finally, interior estimates and in some cases estimates up to plane portions of the boundary are derived for difference quotients of solutions of elliptic difference equations.