Remarks on the Order of Convergence of Discrete Analogs for Second-Order Elliptic Partial Differential Equations

Abstract

IN SOLVING partial differential equations numerically by the method of finite differences, the solution u of the differential equation is replaced by the solution v of a finite set of analogous equations li(v) = si. The difference u v is expected to become smaller with successive mesh refinements. Order of refers to the maximum value of u v I as related to the fineness of the mesh. This paper is concerned with estimating the order of convergence for various discrete analogs of boundary value problems associated with certain elliptic partial differential equations. In particular we examine the effect of interfaces and mesh size changes on the order of convergence for the Dirichlet problem connected with Poisson's equation in two dimensions. The methods used here are based on those used by Gerschgorin [1] and also by Batschelet [11]. These methods give bounds on the error committed by using the discrete analog of the boundary value problem in terms of the derivatives through the fourth order of the true solution, provided they exist and are bounded in the domain. Boundedness of the fourth derivatives is a requirement often not satisfied in even the simplest practical problems, and therefore must be recognized as a heavy limitation on Gerschgorin's method. However, this limitation has been made easier to bear by recent results concerning regularity properties in the closed region of solutions to strongly elliptic equations, for example in [6], [7], [8]. A conclusion is that if the coefficients in equation (1.1) below are in C3, 0 < A < 1, and the Dirichlet data and boundary are in C4, then the solution is in CA4 on the closed region, i.e., the fourth order derivatives of the solution are uniformly Holder continuous with exponent ,u on the closed region. However, a second difficulty is that the magnitude of the bounds on the derivatives is usually unknown even though the existence of such bounds is guaranteed. In the recent works of Walsh, Young, Wasow, and others, these difficulties have been overcome for special equations or special geometry, as in [5]. Nonetheless, Gerschgorin's bounds are useful in that they afford some information concerning the improvement one can expect for a given mesh refinement, as well as an indication of the magnitude of the problem. For most cases of practical interest, it is desirable that the discrete analogs have convergence of order h2, where h is the mesh spacing. Generally such problems may be solved satisfactorily with the order of 103 mesh points, a reasonable number on modern computers. When the order of convergence is only h, many