Adaptive finite element methods for computing band gaps in photonic crystals

In this paper we propose adaptive finite element methods for computing the band structure of 2D periodic photonic crystals and of photonic crystal fibres, modelled as spectral problems for Maxwell’s equations under either TM or TE polarisation. With the application of the Floquet transform the problem of computing the spectrum can be reduced to the computation of the discrete spectra of each member of a family of periodic Hermitian eigenvalue problems on the unit cell, parametrised by the quasimomentum. These eigenvalue problems involve non-coercive elliptic operators with discontinuous coefficients and are solved by adaptive finite elements. We propose an error estimator of residual type and show it is reliable and efficient for each eigenvalue problem in the family. We illustrate the benefits of the resulting adaptive method in practice both for fully periodic structures and also for the computation of eigenvalues in the band gap of structures with defect, using the supercell method. MSC2010 Subject Classification: 65M50, 65M60, 65F15