Convergent Adaptive Finite Element Methods for Photonic Crystal Applications

Abstract

Use policyThe full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes provided that:• a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders.Please consult the full DRO policy for further details. AbstractWe prove the convergence of an adaptive finite element method for computing the band structure of 2D periodic photonic crystals with or without compact defects in both the TM and TE polarization cases. These eigenvalue problems involve non-coercive elliptic operators with discontinuous coefficients. The error analysis extends the theory of convergence of adaptive methods for elliptic eigenvalue problems to photonic crystal problems, and in particular deals with various complications which arise essentially from the lack of coercivity of the elliptic operator with discontinuous coefficients. We prove the convergence of the adaptive method in an oscillation-free way and with no extra assumptions on the initial mesh, beside the conformity and shape regularity. Also we present and prove the convergence of an adaptive method to compute efficiently an entire band in the spectrum. This method is guaranteed to converge to the correct global maximum and minimum of the band, which is a very useful piece of information in practice. Our numerical results cover both the cases of periodic structures with and without compact defects.

An a posteriori error estimator for $$hp$$ -adaptive continuous Galerkin methods for photonic crystal applications

“…Let T λj denote the orthogonal projection of H 1 onto E(λ j ) with respect to the inner product (·, ·) κ,A,B defined in (18). The following lemma is proved in [19,Lemma 3.3].…”

“…The figure confirms that between the first and the second band there is a gap. The surfaces in Figure 3(b) are computed with Algorithm 3 in [18], which is particularly efficient in computing entire bands. As predicted by the theory [23], the presence of a compact defect in the periodic structure can consequently create localized eigenvalues in the gaps that correspond to trapped modes.…”