Figure
1. A portion of a periodic structure

This
project has been done in collaboration with Prof. Ivan
Graham.
We
derived an a posteriori error estimator based on
residuals for eigenvalue problems arising from photonic crystals (PC).
Our goals were: firstly, to compute in an efficient way the gaps in the
spectra of periodic media in both the TE and TM modes. Secondly, we
used the same error estimator to compute efficiently the trapped mode
of photonic crystal fibers with defects.
We proved that our error estimator is reliable and efficient. Moreover,
it is able to capture well the interfaces in the media especially in
presence of high contrast.
Periodic Media:
Figure
2. A solution with singularities in the gradient around the corners


Figure
3.
Another solution
with singularities in the gradient around the corners

In
Fig. 1 we have an example of periodic structure
formed by square inclusions of a different material from the background.
Typical example of solutions for this kind of structure for different
value of the contrast between the two materials are illustrated in Fig.
2 and Fig. 3. The solution could be characterise by singularities in
the gradient around the corners and sharp edges along the interfaces.
Fig.
4 illustrates how the error estimator drives the
mesh adaptivity to refine around the corners of the interface of the
square inclusion, where the singularities in the gradient of the
solutions are situated.
Since the goal is the computation of the spectral gaps of the
structure, we conclude with Fig. 5 showing the spectral bands
surrounding the gaps. To speed up the computation we use a parallel
machine to compute the spectra.
