# PDE eigenvalue iterations with applications in two-dimensional photonic crystals

## TL;DR

This paper develops iterative methods for solving PDE eigenvalue problems in 2D photonic crystals, including nonlinear cases, demonstrating convergence and efficiency improvements through adaptive refinement and Newton-type iterations.

## Contribution

It extends classical eigenvalue algorithms to infinite-dimensional PDE problems and introduces a Newton method for nonlinear eigenproblems in photonic crystal modeling.

## Key findings

- Inverse power method converges on operator level.
- Adaptive mesh refinement accelerates computations.
- Newton-type iteration achieves local quadratic convergence.

## Abstract

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.