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.
Keywords: Nonlinear eigenvalue problem, photonic crystals, inverse power method, Newton iteration
@article{M2AN_2020__54_5_1751_0,
author = {Altmann, Robert and Froidevaux, Marine},
title = {PDE eigenvalue iterations with applications in two-dimensional photonic crystals},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {1751--1776},
year = {2020},
publisher = {EDP Sciences},
volume = {54},
number = {5},
doi = {10.1051/m2an/2020014},
mrnumber = {4126311},
zbl = {1480.65323},
language = {en},
url = {https://www.numdam.org/articles/10.1051/m2an/2020014/}
}
TY - JOUR AU - Altmann, Robert AU - Froidevaux, Marine TI - PDE eigenvalue iterations with applications in two-dimensional photonic crystals JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2020 SP - 1751 EP - 1776 VL - 54 IS - 5 PB - EDP Sciences UR - https://www.numdam.org/articles/10.1051/m2an/2020014/ DO - 10.1051/m2an/2020014 LA - en ID - M2AN_2020__54_5_1751_0 ER -
%0 Journal Article %A Altmann, Robert %A Froidevaux, Marine %T PDE eigenvalue iterations with applications in two-dimensional photonic crystals %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2020 %P 1751-1776 %V 54 %N 5 %I EDP Sciences %U https://www.numdam.org/articles/10.1051/m2an/2020014/ %R 10.1051/m2an/2020014 %G en %F M2AN_2020__54_5_1751_0
Altmann, Robert; Froidevaux, Marine. PDE eigenvalue iterations with applications in two-dimensional photonic crystals. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 54 (2020) no. 5, pp. 1751-1776. doi: 10.1051/m2an/2020014
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