Error Estimates in Elliptic Boundary Value Problems

Abstract

In these lectures, I will discuss some methods of obtaining error estimates for finite difference approximations to solutions of elliptic differential boundary value problems. Because of the limited time, I shall restrict my attention to the Dirichlet problem, although some of the methods are easily carried over to other boundary conditions. The first part will be devoted to second order problems. In fact, in order to illustrate the methods, I will restrict my attention to the Dirichlet problem for Poisson's equation, with “zero boundary values” and the classical Dirichlet problem for Laplace's equation. The last part will be devoted to some results on higher order elliptic equations. In most cases, I will not give more than a sketch of the proof, indicating the details. Instead of choosing to discuss a general class of operators and a corresponding general class of difference schemes, I shall consider a specific operator and certain specific difference schemes so as not to obscure the essential points of the method of analysis.

I will not, during these talks, make extensive references to related work but shall include in the bibliography a number of closely related papers. I shall restrict the references to the specific results under discussion.