The Optimal Relaxation Parameter for the SOR Method Applied to a Classical Model Problem

Abstract

The successive overrelaxation (SOR) method is an example of a classical iterative method for the approximate solution of a system of linear equations. Its iteration matrix depends on a relaxation parameter. There is no explicit formula for the optimal relaxation parameter in terms of properties of the system matrix of a general system matrix. However, for the classical model problem of a finite difference approximation to the Poisson equation, a formula for the optimal relaxation parameter can be derived. Beyond this model problem, this result is also useful as guidance for the choice of the parameter in other problems. This paper presents the detailed derivation of the formula for the optimal relaxation parameter for the model problem and extends the well-known one- and two-dimensional results to higher dimensions