On the Degree of Convergence of Solutions of Difference Equations to the Solution of the Dirichlet Problem

Abstract

1. Introduction. Although finite difference methods are finding frequent use in the solution of boundary value problems associated with partial differential equations of elliptic type, little appears to be known about the accuracy of these methods. The object of the present paper is to study degree of convergence of the difference equation solution of Laplace's equation in the square, as the mesh size approaches zero, under various continuity hypotheses on the boundary values. Gerschgorin [2)2 has given some error bounds but these involve bounds on certain partial derivatives of the solution in the closed region. Finite bounds for these derivatives may not exist and even when they do exist are very hard to find except under very stringent assumptions on the boundary values. The only available error bounds not involving higher derivatives of the solution are the non-explicit bound given by Rosenbloom [10] and the explicit error bounds given by Wasow [13] and by Walsh and Young [12] for the case of the rectangle. In [13] very severe differentiability conditions were imposed. In [10] and [12] uniform error bounds were obtained which are, in general, much too large for a particular point. For instance in [12], the uniform error was shown to vanish of the order h2/7, where h is the mesh size, provided the boundary values satisfy a Lipschitz condition of order one. For this case we now show that the error tends to zero as the first power of h uniformly on any closed interior region.We also obtain other results on the degree of convergence uniformly in any closed interior region of the square expressed in terms of assumed modulus of continuity of the given function, or modulus of continuity except for finite jumps, or modulus of continuity of the derivative, or total variation.Our results can be easily extended to rectangles, and it is hoped that they will give some indication as to the situation to be expected for more general regions. Explicit solutions.As general background for our treatment we state THEOREM 2.1. Let R denote a bounded plane region whose boundary S consists of a finite number of mutually disjoint Jordan curves, and let the function F(x, y) be bounded on S and continuous on S except perhaps for a finite number of points Q. Then there exists a unique function u(x, y) harmonic and bounded in R, and continuous in n = R + S except at the points Q, and equal to F(x, y) on S.If the functions Uk(X, y) harmonic in R are bounded in their totality, if each function is continuous in n except perhaps in the points Q, and if Uk(X, y) approaches

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