Mixed Spectral-Element Methods for 3-D Maxwell’s Eigenvalue Problems With Bloch Periodic and Open Resonators

Two mixed spectral-element methods (MSEMs) are proposed to solve Maxwell’s eigenvalue problems with Bloch (Floquet) periodic and open resonators to remove the dc spurious modes present in the traditional numerical methods. The first MSEM combines the variational form of Gauss’s law to make the electric field satisfy the divergence-free condition. This method does not need to modify the Arnoldi algorithm for the eigenvalue problems, but the number of degrees of freedom (DOFs) is larger than that of the traditional spectral-element method (SEM). The second MSEM, as a generalization of the tree–cotree technique, can eliminate the dc spurious modes by imposing the constrained equations on the Ritz vectors in the Arnoldi algorithm without increasing the DOF from the traditional SEM. We expand the electric field in terms of the edge basis functions associated with the cotree edges and the gradient of the nodal basis functions so that the discrete gradient matrix $\bar {\bar {\mathrm {G}}}$ consists of the identity matrix $\bar {\bar {\mathrm {G}}}_{t}$ and the zero matrix $\bar {\bar {\mathrm {G}}}_{c}$ , making the matrix inversion of $\bar {\bar {\mathrm {G}}}_{t}$ trivial. Finally, in the numerical experiments, we compare the computational costs of the MSEMs and finite-element method (FEM) in COMSOL to illustrate the high efficiency of the MSEMs.