Preconditioning for Regular Elliptic Systems

Abstract

In this paper we examine preconditioning operators for regular elliptic systems of partial differential operators. We obtain general conditions under which the preconditioned systems are bounded. We also provide some useful guidelines for choosing left and right preconditioning operators for regular elliptic systems. The condition numbers of the discrete operators arising from these preconditioned operators are shown to be bounded independent of grid spacing. Several examples of two-dimensional regular elliptic systems are discussed, including scalar elliptic operators and the Stokes operator with several different boundary conditions. Several preconditioners for these regular elliptic systems are presented and used in numerical experiments illustrating the theoretical results.