Band structures calculation of photonic crystals for frequency-dependent materials in the transverse electric (TE) case was recently studied by Xiao et al. (J. Sci. Comput. 87:1–16, 2021). The aim of this paper is to study the transverse magnetic (TM) case. Since the solution regularity is low on the whole domain, the convergence analysis of eigenvalues will be more complex. This model essentially calculates the discrete spectra of a nonlinear eigenvalue problem with periodic boundary conditions. By introducing an elliptic interface source problem, the nonlinear eigenvalue problem is formulated as an eigenvalue problem of a holomorphic Fredholm operator function of index zero, then the finite element methods are used for discretization, and the convergence is proved by the abstract approximation theory for holomorphic operator functions. Finally, we use the spectral indicator method to practically compute eigenvalues. Numerical simulation results are presented to demonstrate the accuracy and the performance of the method.
