Preconditioning conjugate gradient method for nonsymmetric systems

Abstract

It is well known that the preconditioned conjugate gradient algorithms (PCG) work very well (for both symmetric and nonsymmetric problems) if the preconditioned is “good enough”. But, in many cases, “good enough” means that for solving (during the application of (PCG)) the systems in which the preconditioning matrix appears too much computational work is necessary. In this paper we present, for systems arising from finite differences or finite element discretizations, both a method of preconditioning and some possibilities to approximate the exact solutions of the systems in which the preconditioner appears. We present a theoretical study of the influence of these approxiamtions on the convergence behaviour of the (PCG) algorithm. In the last section we present some test on the one-dimensional steady state heat convective transfer problem.