An Adaptive Finite Element Method for Linear Elliptic Problems

We propose an adaptive finite element method for linear elliptic problems based on an optimal maximum norm error estimate. The algorithm produces a sequence of successively refined meshes with a final mesh on which a given error tolerance is satisfied. In each step the refinement to be made is determined by locally estimating the size of certain derivatives of the exact solution through computed finite element solutions. We analyze and justify the algorithm in a model case. Introduction. Recently, adaptive finite element methods for elliptic problems have attracted much interest, see, e.g., [l]-[4], [6], [7], [13], and are rapidly becoming increasingly important in applications. The basic problem concerning such adaptive methods is roughly the following: Given an elliptic problem with no a priori knowledge of the behavior of the exact solution and a finite element method for this problem together with an error tolerance 8 > 0 and a certain norm, construct an automatic procedure for finding a finite element mesh such that the error in the corresponding finite element solution is at most 6 in the given norm. One further requires the constructed mesh to be efficient in the sense that, e.g., the number of elements is nearly minimal. A typical adaptive procedure could be expected to involve a sequence of finite element solutions on successively refined meshes (starting with, e.g., a quasi-uniform mesh), and the procedure would end when the error is smaller than or equal to the given tolerance. At each step of the procedure an estimate of the error on the given mesh would be made, and in case the error tolerance is not met, a refined mesh to be used in the next step would be constructed. Typically, the procedure would generate meshes which are refined in regions where the exact solution is nonsmooth such as, e.g., neighborhoods of corners in a polygonal domain. In the methods proposed by BabuSka and coworkers [2]-[4], the error estimate at each step is based on solving local problems involving a local residual, and the refinements are carried out according to the size of the solutions of the local problems. This method seems to produce reasonable meshes in many cases but appears to be difficult to theoretically justify in several dimensions (cf. [2], [4]). The purpose of this note is to present and analyze, in a model case, an adaptive procedure which is based on a different approach than the BabuSka method. As a model problem we shall consider the Poisson equation { -Au = f in 0, (0.1) I ' { u = 0 onT, Received June 24, 1985; revised October 16, 1986. 1980 Mathematics Subject Classification (1985 Revision). Primary 65N15, 65N30. ©1988 American Mathematical Society 0025-5718/88 $1.00 + $.25 per page