A Novel Matrix-Free Finite Element Method for Time-Harmonic Maxwell's Equations

A frequency-domain finite element method (FEM) with the low-storage matrix-free feature is proposed for efficient analysis of electromagnetic fields. As opposed to conventional frequency-domain FEM which requires the matrix assembly before solving, the proposed matrix-free algorithm avoids the assembly and storage of the global matrix. In this approach, the global sparse matrix-vector (SpMV) multiplication is decomposed into element-wise matrix-vector (MV) multiplications. Additionally, the sum factorization technique is applied on tensorial basis functions to reduce the complexities of local MV multiplications. The numerical results demonstrate the superiority of our algorithm over traditional FEM in terms of both memory and time consumption. Besides, the improvement is more profound when higher-order basis functions are considered. A speedup of more than 10 against matrix-assembled solvers is observed. Given that memory transfer is bounded more than computation resources in modern supercomputer architectures, our algorithm is more friendly to high-performance computing platforms.