Optimum iterative methods for the solution of singular linear systems arising from the discretization of elliptic P.D.E.'s

Abstract

This paper refers to three iterative methods, namely the generalized extrapolated Jacobi (GJOR), the generalized successive overrelaxation (GSOR) and the second order 2-cyclic Chebyshev semi-iterative ones, for the solution of a singular linear system Ax = b, with det( A) = 0 and b in the range of A. As is known, under certain basic conditions (assumptions), one can determine the various parameters involved in the aforementioned methods so that each one of them semiconverges asymptotically as fast as possible. The theory is applied to the singular linear systems arising from the discretization of the 2-dimensional Neumann and periodic boundary value problems for the Poisson equation in a rectangle. After the verification of the validity of the basic assumptions for each method, the optimum parameters are obtained and by means of them the optimum asymptotic semiconvergence rates for the JOR and SOR methods are determined. These are compared with the convergence rates of the Jacobi and the SOR methods used for the solution of the corresponding nonsingular linear system for the Dirichlet model problem for Poisson equation and various conclusions regarding their relative asymptotic behaviors are drawn.