Estimation of the Relaxation Factor for Small Mesh Size

Recently Frankel [1] and Young [3, 4] have developed a method of successive overrelaxation for the solution of the difference equation analogue of the Dirichlet problem for a linear elliptic partial differential equation. The purpose of this note is to present a simple asymptotic theory of the overrelaxation process which depends on the assumption that the mesh size h approaches zero, but which describes adequately the practical aspects of the earlier work. Thus the results discussed are not essentially new, but it is hoped that our approach to them will provide an easy insight into the overrelaxation scheme for those interested in its application to specific examples. We consider the Laplace difference equations for an unknown function u of two independent variables in a region D covered by net points spaced h units apart. Ina standard notation, we use subscripts i and j to refer to the location of net points in our grid and we use the superscript re to indicate steps in the relaxation process, so that, in its simplest form, the method of successive overrelaxation can be described by the equation