A theoretically rigorous solution for fundamentally eliminating the low-frequency breakdown problem in finite-element-based full-wave analysis

In this paper, we develop 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. Hence, the lowfrequency breakdown problem is naturally bypassed. In addition, we found that the zero eigenvalues of the generalized eigenvalue problem cannot be obtained exactly 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 theoretically rigorous, and hence applicable to any frequency across the full electromagnetic spectrum. Not only can it be used to fundamentally eliminate the low-frequency breakdown problem, but also can it be employed to benchmark the accuracy of existing electromagnetic solvers at low frequencies including static solvers.