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Regularity for Maxwell eigenproblems in photonic crystal fibre modelling

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Monique Dauge,
Richard A. Norton (Otago, New Zealand), and
Robert Scheichl (Bath, UK).
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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 $H^{2-\eta}$ for
all $\eta > 0$. This estimate is sharp in the sense that these
components do not belong to $H^2$, in general. However, global
regularity is restricted by the presence of an interface
between these subdomains and the interface conditions imply
only $H^{3/2-\eta}$ 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.

Published online April 2014 doi:10.1007/s10543-014-0487-z

BIT Numerical Mathematics.
**55**, 1, 59-80 (2015)