A fast direct finite element solver for large-scale 3-D electromagnetic analysis

A fast direct solver is developed for the vector finite element based analysis of general 3-D electromagnetic problems. In this solver, we fully take advantage of the sparsity in the original matrix of size N and the reduced fill-ins due to nested dissection ordering. We also organize the overall factorization into a sequence of partial factorizations of dense smaller matrices by the multifrontal algorithm. We then accelerate the computation of all the intermediate dense matrices by ℌ-matrix based fast arithmetic. The computational complexity of the proposed solver is theoretically analyzed and shown to be O(N) in both time and storage for problems with a relatively small electric size. The solver's controlled accuracy is also proved. Numerical experiments demonstrate that the proposed solver outperforms both the state-of-the-art multifrontal solver and the ℌ-based direct solver for 3-D electromagnetic analysis.