Sparse direct matrix solvers of finite element discretizations in electromagnetics

Abstract

The paper outlines some direct solution strategies for sparse matrices arising from the finite element (FE) discretization of electromagnetic (EM) models, and explores whether the conventional thinking followed by highly optimized direct method tools for sparse matrices is indeed the best available option for directly solving FEM problems in EM. Factorization memory and time, solution time, parallelization potential, and error will be considered in the comparisons between approaches. Drawing from advances in the area of integral equation (IE) methods that rely on relatively small but dense matrices, we will conjecture that direct solvers that minimize factorization fill-in while maintaining sparsity may not lead to the best overall strategy after all. We present an direct solution algorithm for FEM, based on domain decomposition that generates and operates on a lesser sparse matrix with block-wise sparse pattern, thus bridging the gap between the two extreme matrix structures, (a large fully sparse encountered in FEM and a smaller fully dense encountered in IE methods). Early results on 3D arbitrary tetrahedron meshes and arbitrary volumetric geometries suggest that the proposed approach significantly outperform state-of-the-art sparse matrix direct solvers in terms of memory while maintaining competitive run times for serial implementations.