Finite Difference Methods for Elliptic Equations

Abstract

The early development of numerical analysis of partial differential equations was dominated by finite difference methods. In such a method an approximate solution is sought at the points of a finite grid of points, and the approximation of the differential equation is accomplished by replacing derivatives by appropriate difference quotients. This reduces the differential equation problem to a finite linear system of algebraic equations. In this chapter we illustrate this for a two-point boundary value problem in one dimension and for the Dirichlet problem for Poisson’s equation in the plane. The analysis is based on discrete versions of the maximum principles of the previous two chapters.