An a posteriori error estimator for hp-adaptive discontinuous Galerkin methods for computing band gaps in photonic crystals

In this paper we propose and analyze an hp-adaptive discontinuous finite element method for computing the band structure of 2D periodic photonic crystals. The spectrum of a 2D photonic crystal is approximated by computing the discrete spectrum of members of a family of periodic Hermitian eigenvalue problems on the primitive cell, parametrized by a two-dimensional parameter - the quasimomentum. We propose a residual-based error estimator and show that it is reliable and efficient for all eigenvalue problems in the family. In particular we prove that if the error estimator converges to zero, then the distance of the computed eigenfunction from the true eigenspace also converges to zero, and so the computed eigenvalue converges to a true eigenvalue. The results hold for eigenvalues of any multiplicity. We illustrate the benefits of the resulting hp-adaptive method numerically, both for fully periodic crystals and also for crystals with defects.