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Adaptive finite element methods for computing band gaps in photonic crystals. (English) Zbl 1241.82077

Summary: We propose and analyse adaptive finite element methods for computing the band structure of 2D periodic photonic crystals. The problem can be reduced to the computation of the discrete spectra of each member of a family of periodic Hermitian eigenvalue problems on a unit cell, parametrised by a two-dimensional parameter – the quasimomentum. These eigenvalue problems involve non-coercive elliptic operators with generally 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. 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 with double the rate. We also prove that, if the distance of a computed sequence of approximate eigenfunctions from the true eigenspace approaches zero, then so must the error estimator. The results hold for eigenvalues of any multiplicity. 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.

MSC:

82D25 Statistical mechanics of crystals
82-08 Computational methods (statistical mechanics) (MSC2010)
78A25 Electromagnetic theory (general)
78M10 Finite element, Galerkin and related methods applied to problems in optics and electromagnetic theory
35Q61 Maxwell equations
78A60 Lasers, masers, optical bistability, nonlinear optics
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N25 Numerical methods for eigenvalue problems for boundary value problems involving PDEs

Software:

ARPACK
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References:

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