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Performance analysis of the H-matrix-based Fast Direct Solver for finite-element-based analysis of electromagnetic problems

A finite-element method (FEM) based analysis of a complex electromagnetic problem generally results in a large-scale system matrix. Although the matrix is sparse, solving it can be a computational challenge when the problem size is large. A direct solution can be computationally intensive. In [1], the optimal operation count of the direct solution of a general FEM matrix was shown to be O(N1.5). State-of-the-art finite-element-based solvers rely on iterative approaches to solve the large-scale system matrix. Compared to iterative solvers, direct solvers have advantages when the number of iterations is large or the number of right hand sides is large. In [2], an H-matrix-based [3] fast direct solver was developed for the finite element based analysis of electromagnetic problems. It was shown that although the inverse of a finite element matrix is generally dense, it can be stored in O(NlogN) units, and computed in O(Nlog2N) complexity. The numerical scheme is applicable to arbitrary structures and problems. The theoretical analysis of complexity and accuracy of the H-matrix-based direct FEM solver was not given in [2] due to space constraint. In this paper, we give a detailed complexity and error analysis of the H-matrix-based fast direct FEM solver for electromagnetic analysis.