Preconditioned iterative methods for partial differential equations

In this paper we consider several preconditioners and iterative methods for solving the linear algebraic system associated with a partial differential equation. Our interest stems from earlier work in Method of Lines (MOL) software for solving kinetics-diffusion equations and a recognition that the solution of the underlying linear system at each timestep is crucial in terms of computational storage and time. We are interested in developing an approach to handle nonsymmetric matrices so that we can deal with convective terms in the partial differential equation (PDE). To examine our methods we consider a model problem which has been used in related work. With regard to the approach there are two aspects: the preconditioner and the iterative method. Among the preconditioners considered are normal form LU factorization and variations related to approximate inverses. The iterative methods include normal form conjugate gradients and related nonsymmetric methods (ORTHOMIN and ORTHODIR). We have found that the use of either an approximate LU factorization or an approximate inverse in combination with normal form conjugate gradient iteration provides an effective approach for solving our model problem. This result suggests potential use of approximate inverses for parallel computation. 5 refs., 4 figs.