  We consider PDE eigenvalue problems as they occur in two-dimensional photonic
crystal modeling. If the permittivity of the material is frequency-dependent,
then the eigenvalue problem becomes nonlinear. In the lossless case,
linearization techniques allow an equivalent reformulation as an extended but
linear and Hermitian eigenvalue problem, which satisfies a Garding inequality.
For this, known iterative schemes for the matrix case such as the inverse power
or the Arnoldi method are extended to the infinite-dimensional case. We prove
convergence of the inverse power method on operator level and consider its
combination with adaptive mesh refinement, leading to substantial computational
speed-ups. For more general photonic crystals, which are described by the
Drude-Lorentz model, we propose the direct application of a Newton-type
iteration. Assuming some a priori knowledge on the eigenpair of interest, we
prove local quadratic convergence of the method. Finally, numerical experiments
confirm the theoretical findings of the paper.
