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 a hp-adaptive discontinuous finite element method for computing the band structure of 2D periodic photonic crystals. The problem can be reduced to the computation of the discrete spectrum of each member in a family of periodic Hermitian eigenvalue problems on the primitive cell, parametrised 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 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 in practice, both for fully periodic crystals and also for the computation of eigenvalues in the band gaps of crystals with defects