Keywords: finite element method, sparse linear systems, normalized approximate inverses, preconditioning, parallel preconditioned conjugate gradient method, parallel generalized minimum residual method, distributed computations.

Matrix computations are of central importance in many engineering and scientific problems. Over the last decades the need for high performance computing has had an important effect on the design of modern computer systems. Hence research efforts were focused on the production of parallel computational methods for solving systems of linear equations. Such research work has concentrated on distributed memory systems.

Many of the well-known preconditioning methods are highly sequential and difficult to implement on parallel computers. Recently, the derivation of parallel numerical algorithms was the main objective for which several forms of approximate inverses of a given matrix, based on adaptive approximate factorization procedures have been proposed. The main motive for the derivation of the approximate inverse lies in the fact that they can be efficiently used in conjunction with explicit preconditioned iterative schemes which are appropriate for solving sparse finite element linear systems on distributed memory systems. The effectiveness of the explicit approximate inverse preconditioning schemata relies on the construction and use of efficient preconditioner factors in the sense that the preconditioners are close approximants to the coefficient matrix and are fast to compute in parallel. Since the normalized explicit preconditioned iterative schemes are the computationally demanding part, there is a need to parallelize such methods for distributed memory systems, in order to achieve better performance.

The purpose of this work is the derivation of normalized explicit preconditioned iterative schemes, based on a class of normalized approximate inverse matrix techniques for finite element symmetric matrices, for the efficient solution of sparse linear systems. The normalized explicit preconditioned conjugate-gradient (NEPCG) and the normalized explicit preconditioned generalized minimum residual (NEPGMRES) methods are considered and parallelized for distributed memory systems using the message passing interface (MPI) communication library.

The performance and applicability of the proposed methods on a three-dimensional boundary value problem are discussed and numerical results are given for uniprocessor and distributed memory systems, using MPI on a cluster of thirty-two processors.

The main advantage of the proposed methods is that the normalized approximate inverse is computed explicitly and can be efficiently used in conjunction with the parallel normalized explicit preconditioned iterative methods, which were implemented with the MPI communication library. For the proposed schemes, good parallel results have been obtained, indicating the qualitative agreement with the theoretical estimates.

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