Resonance problems arise in many fields of research. An example are photonic crystals, in which the propagation of waves is defined by resonances. Photonic crystals with band gaps are of special interest. Band gaps are regions of frequencies which cannot propagate through the crystal. To calculate the band structure of photonic crystals many linear resonance problems need to be solved. Fast and reliable linear eigenvalue solvers are needed. Lately, metallic photonic crystals have become more interesting. Differently from photonic crystals the electric permittivity of metallic photonic crystals depends on the frequency leading to rational resonance problems. These problems come with a high computational cost. In this thesis, we introduce an efficient eigenvalue solver for large rational eigenvalue problems. At first, the resonance problems for two and three dimensional metallic photonic crystal are derived from Maxwells equations. Then, they are discretised with Bloch periodic high order finite elements in Netgen/NGSolve. The arising large rational matrix eigenvalue problems are linearised with a rational linearisation schema and solved by the shiftandinvert Arnoldi method. By combining linearisation with the shiftandinvert Arnoldi, systems of linear equations with dimensions larger than the original matrix size have to be solve in each iteration. With the introduced rational linearisation these large systems of linear equations can be reduced to the original problem size. The proposed combination of these two algorithms is applied to two and three dimensional metallic photonic crystals and compared to the shiftandinvert Arnoldi with a standard polynomial linearisation. The appearance of plasmon frequencies is witnessed and the influence on the solver is studied. We show that the proposed method is a reliable and fast solver for large rational eigenvalue problems.
