Large scale finite element computations with GMRES-like methods on a Cray Y-MP (Q1917402)

The author is interested in methods for solving large non-symmetric systems such as arise from finite element discretization of convection-diffusion equations. These methods should be appropriate for implementation in a commercial finite element computer program such as DIANA. Methods of interest in the present paper include GMRES, GCR, and GMRESR with various preconditioning strategies. The author presents several approaches for optimizing the implementation of matrix-vector products on vector and parallel computer architectures, concentrating on elemental methods rather than ones which require explicit construction of the stiffness matrix. Element-by-element preconditioners are also discussed. The discussion is heavily influenced by the architecture of vector computers such as the Cray Y-MP, although other computers are mentioned. The author presents numerical experiments performed with two non-trivial convection-diffusion equations. GMRES, GCR, and GMRESR methods are compared with several different preconditioning strategies. There is no decisive victor in the comparison, but GMRESR emerges as a very attractive candidate.