A Theoretically Rigorous Full-Wave Finite-Element-Based Solution of Maxwell's Equations From dc to High Frequencies

It has been observed that finite element based solutions of full-wave Maxwell's equations break down at low frequencies. In this paper, we present a theoretically rigorous method to fundamentally eliminate the low-frequency breakdown problem. The key idea of this method is that the original frequency-dependent deterministic problem can be rigorously solved from a generalized eigenvalue problem that is frequency independent. In addition, we found that the zero eigenvalues of the generalized eigenvalue problem cannot be obtained as zeros because of finite machine precision. We hence correct the inexact zero eigenvalues to be exact zeros. The validity and accuracy of the proposed method have been demonstrated by the analysis of both lossless and lossy problems having on-chip circuit dimensions from dc to high frequencies. The proposed method is applicable to any frequency. Hence it constitutes a universal solution of Maxwell's equations in a full electromagnetic spectrum. The proposed method can be used to not only fundamentally eliminate the low-frequency breakdown problem, but also benchmark the accuracy of existing electromagnetic solvers at low frequencies including static solvers. Such a benchmark does not exist yet because full-wave solvers break down while static solvers involve theoretical approximations.