A Lowest-Order Mixed Mortar-Element Method for 3-D Maxwell’s Eigenvalue Problems With the Absorbing Boundary Condition

Combining the traditional mymargin mortar-element mymargin method (MEM) and the tree–cotree technique, we present a mixed MEM (MMEM) based on a hybrid mesh to solve 3-D Maxwell’s eigenvalue problems with the absorbing boundary condition (ABC). In this MMEM, to couple the different discretizations in hexahedral and tetrahedral subdomains, we design a mortar condition by which the degrees of freedom (DOFs) associated with the interface of hexahedral subdomains can be eliminated. When implementing the tree–cotree technique, the electric field and the test function are expressed by the lowest-order vector basis functions (edge elements) and the gradient of the first-order scalar nodal basis functions, which leads to a singular submatrix $\bar {\bar {E}}_{\mathrm {sub}}$ in the mortar condition. Therefore, in this work, we introduce a method for selecting tree edges and cotree edges of hexahedral meshes to make $\bar {\bar {E}}_{\mathrm {sub}}$ nonsingular. Numerical experiments show that the MMEM can not only remove dc spurious modes but also retain physical modes. The numerical results of the MMEM are consistent with those of the commercial software COMSOL.

I. Introduction

The tangential vector finite-element method (FEM) is a powerful tool for solving Maxwell’s eigenvalue problems. This method usually matches a purely hexahedral mesh or a purely tetrahedral mesh used to discretize geometric structures. For simplicity, FEM-hex and FEM-tet are introduced to denote the FEM based on the purely hexahedral mesh and the FEM based on the purely tetrahedral mesh, respectively. The advantages and disadvantages of the FEM-hex and the FEM-tet are as follows. Since complex models can be flexibly discretized by the tetrahedral mesh [1], [2], the FEM-tet is often used in the simulation software (e.g., COMSOL Multiphysics and high frequency structure simulator). However, the mass matrix in the FEM-tet is not diagonal, which is not conducive to obtaining the inverse of the mass matrix. On the other hand, although it is more difficult to mesh complex geometric structures with hexahedral elements, the mass matrix in the FEM-hex is very sparse. In particular, when the numerical quadrature points are taken as the interpolation points of the first-order Gauss–Lobatto–Legendre polynomial [3], the mass matrix in the FEM-hex based on the lowest-order basis functions (i.e., incomplete first-order basis functions [4]) can be approximated as a diagonal matrix [5]. Therefore, Maxwell’s eigenvalue problems are transformed into the standard eigenvalue problems [6], which can reduce computational costs.