Parallel Domain Decomposition Preconditioning for the Adaptive Finite Element Solution of Elliptic Problems in Three Dimensions

Abstract

A novel weakly overlapping two level additive Schwarz domain decomposition preconditioning algorithm is presented which is appropriate for the parallel finite element solution of elliptic partial differential equations in three dimensions. This algorithm allows each processor to be assigned one or more subdomains and, as with most parallel domain decomposition solvers, each processor is able to solve its own subproblem(s) concurrently. The novel feature of the technique proposed here is that it requires just a single layer of overlap in the elements which make up each subdomain at each level of refinement, and it is shown that this amount of overlap is sufficient to yield an optimal preconditioner. The number of elements in this overlap region between subdomains is O(h-2 ) as the mesh size h -> 0. This is an improvement over the O(h-3) overlapping elements required to obtain optimality for a conventional two level additive Schwarz algorithm. The quality and effectiveness of this new algorithm is examined using both global uniform and local non-uniform refinement with two representative partitions of the domain . This preconditioning algorithm is then generalized such that the resulting preconditioner is not only suitable for symmetric problems but also for nonsymmetric and convection-dominated elliptic problems. This generalization, in the absence of theoretical or mathematical background, is based on empirical observations. Moreover, it turns out to be more effective and robust than the original symmetric preconditioning algorithm when applied to symmetric positive definite problems. This generalized algorithm is tested on symmetric, nonsymmetric and convection-dominated partial differential equations, where the number of iterations observed suggests that the preconditioner may in fact be optimal, i.e. the condition number of the preconditioned systems is bounded as the mesh is refined or the number of subdomains is increased. Due to non-physical oscillations in the solution of convection-dominated problems when discretized by the Galerkin finite element method, unless the size of elements is sufficiently small, we have extended our implementation of the generalized preconditioning algorithm to be applicable to systems arising from a more stable finite element discretization technique based upon streamline diffusion. Numerical experiments for a wide variety of problems are included to demonstrate the optimal or near-optimal behaviour and quality of this generalized algorithm. Parallel performance of the generalized preconditioning algorithm is also evaluated and analyzed. All the timings quoted are for a SG Origin 2000 computer and all software implementations described in this thesis have been coded and tested using ANSI C and the MPI communication library