PDE eigenvalue iterations with applications in two-dimensional photonic crystals

¹ Institut für Mathematik, Universität Augsburg, Universitätsstr. 14, 86159 Augsburg, Germany
² Institut für Mathematik MA4-5, Technische Universität Berlin, Straß e des 17. Juni 136, 10623 Berlin, Germany

^* Corresponding author: robert.altmann@math.uni-augsburg.de

Received: 3 May 2019
Accepted: 26 February 2020

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 Gårding 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.