Edge elements: from Maxwell equations to photonic crystals
Lucia Gastaldi
Dipartimento di Matematica
Universita di Brescia, via Valotti 9 Brescia, Italy.
Photonic crystals are periodic structures composed of dielectric materials. The
reason for the increase of the interest in this subject is that the spectrum of
the Maxwell operator for such media is expected to have gaps. The presence of
gaps means that there are prohibited frequencies of propagation of
electromagnetic waves going through such crystals. This fact has many potential
applications, for example, in optical communications, filters, lasers and
microwaves.
We analyze a finite element method based on a modification of
Nedelec edge elements. We prove convergence of approximated eigensolutions
to those of the continuous problem under general assumptions on the mesh.
The main result requires minimal regularity hypotheses on the coefficients.