Nonlinear spectral problems frequently occur when damping effects or frequency dependencies are included in a model. Many physically relevant spectral problems induce symmetries in the spectrum. Numerical methods that ignore structure often break these qualitatively important symmetries, which may lead to physically meaningless or unstable results. The importance of preserving eigenvalue symmetries of a matrix polynomial has been discussed in recent years. However, the whole chain, from spectral symmetries of the non-linear problem on an infinite-dimensional space to the numerically calculated eigenvalues has not been studied. The aim of this project is to analyze all steps in the chain and to use the new results to develop highly effective numerical software for calculations of plasmon resonances and band gaps in dielectric and metallic photonic crystals. Such crystal structures that are designed for controlling propagation of electromagnetic waves have numerous applications including telecommunication, solar cells, integrated circuits, and possibly in the future optical or plasmonic computers.
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