On sparse and compact preconditioned conjugate gradient methods for partial differential equations

This paper generalises the preconditioning techniques, introduced by Evans [2], and defines sparse and compact preconditioned iterative methods for the numerical solution of the linear system Au = b.The difference between the methods is shown to depend on whether a conditioning matrix R consists of components derived from a splitting or factorization of A. Some theoretical results for the iterative schemes are given when A has particular properties such as consistent ordering, irreducibility, diagonal dominance, positive definiteness, etc., when derived from the finite difference discretisation of a 2nd order self-adjoint elliptic partial differential equation. Finally, the application of both forms of preconditioning to the Conjugate Gradient method is presented and computational results compared.