H-Matrix-Based Fast Direct Finite Element Solver for Large-Scale Electromagnetic Analysis

In this work, we prove that the sparse matrix resulting from a finite-element-based analysis of electrodynamic problems can be represented by an -matrix without any approximation, and the inverse of this sparse matrix has a data-sparse matrix approximation with error well controlled. Based on this proof, we develop an -matrix-based direct finite-element solver of O(kNlogN) memory complexity and O(kNlogN) time complexity for solving electromagnetic problems, where k is a small variable that is adaptively determined based on accuracy requirements, and N is the number of unknowns. Both inverse-based and LU-based direct solutions are developed. The LU-based solution is further accelerated by nested dissection. Both theoretical analysis and numerical experiments have demonstrated the accuracy and almost linear complexity of the proposed solver in large-scale electrostatic and electrodynamic applications involving over 1 million unknowns. A comparison with the state-of-the-art direct finite element solver that employs the most advanced sparse matrix solution has shown clear advantages of the proposed solver. In addition, the proposed solver is applicable to arbitrarily-shaped threedimensional structures and arbitrary inhomogeneity.