Three Numerical Eigensolvers for 3-D Cavity Resonators Filled With Anisotropic and Nonconductive Media

TL;DR

This paper compares three numerical eigensolvers for 3-D cavity resonator problems with anisotropic, nonconductive media, highlighting their advantages, limitations, and conditions for avoiding spurious modes.

Contribution

It introduces and analyzes three eigensolvers—penalty, augmented, and projection methods—for solving Maxwell's eigenvalue problems in anisotropic media, proving the absence of spurious modes under certain conditions.

Findings

Augmented method is free of spurious modes if the material is not magnetic lossy.

Projection method based on SVD cannot introduce spurious modes.

Numerical experiments verify the theoretical advantages of the proposed methods.

Abstract

This paper mainly investigates the classic resonant cavity problem with anisotropic and nonconductive media, which is a linear vector Maxwell's eigenvalue problem. The finite element method based on edge element of the lowest-order and standard linear element is used to solve this type of 3-D closed cavity problem. In order to eliminate spurious zero modes in the numerical simulation, the divergence-free condition supported by Gauss' law is enforced in a weak sense. After the finite element discretization, the generalized eigenvalue problem with a linear constraint condition needs to be solved. Penalty method, augmented method and projection method are applied to solve this difficult problem in numerical linear algebra. The advantages and disadvantages of these three computational methods are also given in this paper. Furthermore, we prove that the augmented method is free of spurious modes as long as the anisotropic material is not magnetic lossy. The projection method based on singular value decomposition technique can be used to solve the resonant cavity problem. Moreover, the projection method {cannot} introduce any spurious modes. At last, several numerical experiments are carried out to verify our theoretical results.