On the Solution of Poisson's Difference Equation

Two standard principles have previously been used in proving that the solution of an elliptic differential equation can be approximated arbitrarily well by replacing the differential equation by a related difference equation on a sufficiently fine net and then solving the corresponding system of linear algebraic equations. First, for the most important homogeneous equations, e.g., Laplace's equation, one can make use of their character as Euler equations related to certain variational problems (see [1], for example) and thus gain an essential advantage from their particular simplicity; this simplicity is exemplified in the solutions by the fact tha t they have derivatives of any order. When an estimate of the truncation error (i.e., of the difference of the two solutions) is known, a second principle frequently is applied; this is based upon a proof tha t the "error" vanishes in proportion to a certain power of the mesh constant. The applicability of this principle is not limited to the simplest equations, but it indeed requires tha t certain derivatives of the solution exist and are bounded, since the proportionality coefficient usually depends upon the magnitude of these derivatives (see [2]). Neither one of these principles can be used to t reat an inhomogeneous equation whose inhomogeneous term is so irregular tha t the derivatives of the solution function which are needed either do not exist or are not bounded. The simplest example of this kind is Poisson's equation with a discontinuous inhomogeneous term. A third, general method for estimating the difference between the corresponding solutions of differential and difference equations can be based on the use of Green's functions for both equations. The difference of the two Green's functions depends just on the region in question and on the parameter point and variable point. Now, by expressing, respectively, the solutions of the inhomogeneous equations in integral form and as a sum, the error can be estimated by means of Green's functions and the inhomogeneous terms without irrelevant assumptions concerning the differentiability of the inhomogeneous terms. The main difficulty in applying this principle is in obtaining expressions for Green's functions; however, it is possible to construct such functions for regions composed of a finite number of rectangles. The purpose of this paper is, first of all, to illustrate the third method men-