An Organization of Sparse Gauss Elimination for Solving PDEs on Distributed Memory Machines

A computational arrangement of Gauss elimination is presented for solving sparse, nonsymmetric linear systems arising from partial differential equation (PDE) problems. It is particularly targeted for use on distributed memory message passing (DMMP) multiprocessor computers and it is presented and analyzed in this context. The objective of the algorithm is to exploit the sparsity (i.e., reducing computation, communication and memory requirements) and to optimize the data structure manipulation overhead. The algorithm is based on the nested dissection approach, which starts with a large set of very sparse, completely independent subsystems and progresses in stages to a single, nearly dense system at the last stage. The computational efforts of each stage are roughly equal (almost exactly equal for model problems), yet the data structures appropriate for the first and last stages are quite different. Thus we use different types of data structures and algorithm components at different stages of the solution. The new organization is a combination of previous techniques including nested dissection, implicit block factorization, domain decomposition, fan-in, fan-out, up-looking, downlooking, and dynamic data structures.