Domain decomposition techniques for the parallel solution of nonsymmetric systems of elliptic boundary value problems

Parallel block-preconditioned domain-decomposed Krylov methods for sparse linear systems are described and illustrated on large two-dimensional model problems and Jacobian matrices from different stages of a nonlinear multicomponent problem in chemically reacting flows. The main motivation of the work is to examine the practicality of parallelization, under the domain decomposition paradigm, of the solution of systems of equations typical of implicit finite difference applications from fluid dynamics. Such systems presently lie beyond the realm of most of the theory for domain-decomposed symmetric or nonsymmetric scalar operators. We describe techniques depending formally only on the sparsity structure of the linear operator and thus of broad applicability. Results of tests run on an Encore Multimax with up to 16 processors demonstrate their utility in the coarse-granularity parallelization of hydrocodes: parallel efficiencies in the 40 to 100 percent range are available on the largest number of processors employed over a mix of problems, relative to a serial approach employing the same iterative technique (GMRES) and preconditioner (ILU) on a single domain. These efficiencies are already competitive with results from undecomposed parallel implementations of ILU-preconditioned GMRES on the same multiprocessor, and many avenues for their improvement remain unexplored.