In the last decades, mesh adaptivity has been widely used to improve the accuracy
of numerical solutions to many scientific problems. The basic idea is to refine the
mesh only where the error is high, with the aim of achieving an accurate solution
using an optimal number of degrees of freedom. There is a large amount of numerical
analysis literature on adaptivity, in particular on reliable and efficient a posteriori error
estimates. Recently, the question of convergence of adaptive methods has received
intensive interest and a number of convergence results for the adaptive solution of
boundary value problems have appeared.
We proved the convergence of an adaptive linear finite element algorithm for com-
puting eigenvalues and eigenvectors of scalar Hermitian elliptic partial differential op-
erators with discontinuous coefficients in bounded polygonal or polyhedral domains,
sub ject to Dirichlet boundary data or to periodic boundary conditions. Such problems
arise in many applications, e.g., resonance problems, nuclear reactor criticality, and the
modelling of photonic band gap materials, to name but three.
Our refinement procedure is based on two locally defined quantities, firstly, a stan-
dard a posteriori error estimator, and secondly a measure of the variability (or “oscil-
lation”) of the computed eigenfunction. Our algorithm performs local refinement on
all elements on which the minimum of these two local quantities is sufficiently large.
We proved that the adaptive method converges provided the initial mesh is sufficiently
fine. The latter condition, while absent for adaptive methods for linear symmetric
elliptic boundary value problems, commonly appears for nonlinear problems and can
be thought of as a manifestation of the nonlinearity of the eigenvalue problem.
We are particularly interested in applying our converging method to photonic crys-
tal fibers, which are used for optical communications, filters, lasers, switchers and
optical transistors. More specifically we used our convergence adaptive finite element
methods to localize the spectral band gaps of fibers and also to compute accurately
and efficiently the modes trapped in the defects. These modes are very important in
applications because they decay exponentially away from the defects and so they can
travel in the fibers with almost no losses.