Mixed Nitsche’s Method for Maxwell’s Eigenvalue Problems

Abstract

Combining the traditional Nitsche’s method (NM) and Gauss’ law, we propose a mixed Nitsche’s method (MNM) based on a hybrid tetrahedron–hexahedron mesh. With this method, geometrical models can be meshed flexibly and zero-frequency (dc) spurious modes can be effectively removed when solving Maxwell’s eigenvalue problems. The number of degrees of freedom (DOFs) in the MNM remains the same as that in the NM by using the tree–cotree technique. Moreover, we summarize the rules of selecting tree edges for high-order hierarchical basis functions and derive the constrained equations with discrete gradient matrices to suppress Ritz vectors in the Arnoldi algorithm. The MNM is not only suitable for a hybrid mesh but also for purely hexahedral and purely tetrahedral meshes, and there is no need to calculate the discrete gradient matrices and to modify the Arnoldi algorithm. Through numerical experiments, it is found that the MNM converges exponentially for a cube cavity discretized by a hybrid mesh and its efficiency is about four times higher than the commercial software COMSOL.