PDE Eigenvalue Iterations with Applications in Two-dimensional Photonic Crystals

The first part of this paper is devoted to the approximative solution of linear and Hermitian eigenvalue problems where the differential operator satisfies a Garding inequality. For this, known iterative schemes for the matrix case such as the inverse power or Arnoldi method are extended to the infinite-dimensional case. This formally allows one to apply different spatial discretizations in each iteration step and thus, justifies the use of adaptive methods. The second part considers eigenvalue problems as they appear in two-dimensional models of photonic crystals for the computation of band-gaps. If the permittivity of the material is frequency-dependent, then this leads to a nonlinear eigenvalue problem. For this, we consider two strategies. First, a linearization combined with the application of the inverse power method and second, a direct application of Newton's iteration