Convergent adaptive finite element methods for photonic crystal applications

Giani, Stefano

Abstract

We 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.

Citation

Giani, S. Convergent adaptive finite element methods for photonic crystal applications. Manuscript submitted for publication

• Journal Article Type: Article
• Deposit Date: Jan 11, 2011
• Peer Reviewed: Not Peer Reviewed
• Public URL: https://nottingham-repository.worktribe.com/output/1012632
• Additional Information: Preprint of an article submitted for consideration in Mathematical Models and Methods in Applied Sciences (M3AS) © 2010 copyright World Scientific Publishing Company. http://www.worldscinet.com/m3as/mkt/editorial.shtml