Regularity for Maxwell eigenproblems in photonic crystal fibre modelling

M. Dauge, R. A. Norton, R. Scheichl

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The convergence behaviour and the design of numerical methods for modelling the flow of light in photonic crystal fibres depend critically on an understanding of the regularity of solutions to time-harmonic Maxwell equations in a three-dimensional, periodic, translationally invariant, heterogeneous medium. In this paper we determine the strength of the dominant singularities that occur at the interface between materials. By modifying earlier regularity theory for polygonal interfaces we find that on each subdomain, where the material in the fibre is constant, the regularity of in-plane components of the magnetic field are H2−η for all η>0 This estimate is sharp in the sense that these components do not belong to H2, in general. However, global regularity is restricted by the presence of an interface between these subdomains and the interface conditions imply onlyH3/2-η regularity across the interface. The results are useful to anyone applying a numerical method such as a finite element method or a planewave expansion method to model photonic crystal fibres or similar materials.

Original languageEnglish
Pages (from-to)59-80
Number of pages22
JournalBIT Numerical Mathematics
Issue number1
Early online date23 Apr 2014
Publication statusPublished - Mar 2015


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