In this work, we prove that the sparse matrix resulting from a finite-element-based analysis of electrodynamic problems can be represented by an H-matrix without any approximation, and the inverse of this sparse matrix has a data-sparse H-matrix approximation with error well controlled. Based on this proof, we develop an H-matrix-based direct finite-element solver of O(kNlogN) memory complexity and O(k2Nlog2N) 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 inversebased 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 three-dimensional structures and arbitrary inhomogeneity.
Finite Element Methods, Electromagnetic Analysis, Fast Solvers, Direct Solution, H-Matrix, Nested Dissection
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