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.