Project DOT3:
Adaptive Finite Element Methods for nonlinear parameter dependent eigenvalue problems in photonic crystals
Duration: 
June 2014  May 2017 
Project leaders: 
V. Mehrmann, A. Międlar 
Department of Mathematics Technical University of Berlin Strasse des 17. Juni 136 10623 Berlin, Germany 

Tel: +49 (0)30  314 25 736 (office) /  314 21 264  
email:  
University of Kansas Department of Mathematics 1460 Jayhawk Blvd., 405 Snow Hall Lawrence, KS 660457594, USA 

Tel: +1 7858645166 (office)  
email:  
Responsible: 
M. Froidevaux, (formerly: C. Gamst and R. Altmann) 
Department of Mathematics, Technical University of Berlin, Strasse des 17. Juni 136, 10623 Berlin, Germany 

Tel: +49 (0)30  314 79 177 (office)  
email:  
Cooperations: 
Matheon projects D26, BMI2, earlier projects: C22, C29, C33 
External cooperations: 
DFG project A. Międlar D. Kressner (EPF Lausanne) C. Engström (Umeå Universitet) E. Jarlebring (KTH Stockholm) L. Grubišić (University of Zagreb) T. Fukaya (Hokkaido University) W.W. Lin (NCTU Taiwan) Y. Nakatsukasa (University of Oxford) 
Support: 
Einstein Center for Mathematics (ECMath) 
ECMath project website: 
Project OT3 
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Background
Photonic crystals are periodic materials that affect the propagation of electromagnetic waves. These materials occur in nature (e.g. on butterfly wings) and can also be designed and manufactured with certain properties affecting the propagation of electromagnetic waves in the visible spectrum, hence the name photonic crystals. The most interesting (and useful) property of such periodic structures is that for certain geometric and material configurations gaps in the bands of possibly propagating wavelengths can occur. These gaps charaterize intervals of wavelengths that cannot propagate in the periodic structure. Therefore, finding materials and geometries with especially wide bandgaps is an ongoing research effort.
Mathematically, finding such bandgaps for different configurations of materials and geometries can be modelled as a PDE eigenvalue problem with the frequency (or wavelength) of the electromagnetic field as the eigenvalue. These eigenvalue problems depend on various parameters and may be nonlinear in the eigenvalue. The parameters describing the material of the structure are typically nonlinear functions of the searched frequency. The configuration of the periodic geometry may also be modified and can be considered a parameter. Finally, through the mathematical treatment of the PDE eigenvalue problem another parameter, the quasimomentum, is introduced in order to reduce the problem from an infinite domain to a family of problems, parametrised by the quasimomentum, on a finite domain. These are more accurately solvable.
In order to solve the problem of finding a material and geometric structure with an especially wide bandgap, one needs to solve many of those nonlinear eigenvalue problems during each step of the optimisation process. Therefore, it is essential to have an efficient way of solving these eigenvalue problems. Finding such efficient solvers is one aim of this research project. It is known that an efficient way of discretizing PDE eigenvalue problems on geometrically complicated domains is an adaptive Finite Element method (AFEM). To investigate the performance of AFEM for the described problems reliable and efficient error estimators for nonlinear parameter dependent eigenvalue problems are needed.
Solving the finite dimensional nonlinear problem resulting from the AFEM discretization in general cannot be done directly, as the systems are usually large, and thus produces another error to be considered in the error analysis. Another goal in this research project is therefore to equilibrate the errors and computational work between the discretization and approximation errors of the AFEM and the errors in the solution of the resulting finite dimensional nonlinear eigenvalue problems.
Research outline
In order to design adaptive finite element methods to calculate photonic bandgaps we plan to derive a priori and a posteriori error estimates for eigenvalues and eigenfunctions of nonlinear PDE eigenvalue problems (NLEVPs) with the structures encountered in photonic crystal modelling. We will extend the theoretical framework of functional perturbation analysis for the nonlinear eigenvalue problem in variational formulation (Voss 09) and the techniques for a posteriori error estimation in general nonlinear problems, see e.g. (Bänsch, Siebert 95), (Verfürth 96) to the above described cases. We will also investigate to which extent a posteriori error estimates for the nonlinear EVP with nonlinearity in the eigenfunction (Cancès, Chakir, Maday 10), (Chen, He, Zhou 11) and recent results for nonlinear diffusion equations (Ern, Vohralík 13) can be employed.
For polynomial eigenvalue problems the notion of invariant pairs was introduced in (Beyn, Thümmler 09), (Betcke, Kressner 11). The extension of these concepts to general nonlinear EVPs is still under investigation. For a generic case, a novel method exploiting the idea of invariant pairs was introduced in (Beyn 12), (Beyn, Effenberger, Kressner 11). We plan to develop a similar approach to follow frequency paths for the NLEVPs in bandgap computations.
back to topHighlights from DOT3 (20142015)
For the twodimensional model problem described above, a priori error estimates for general nonlinear eigenvalue problems from the ongoing work of A. Międlar and D. Kressner can be applied in the case of frequency dependent electric permittivity functions.
In the case of permittivity functions which are analytic in the frequency, a posteriori error estimators as described in an upcoming technical report (2015) can be derived using the general framework of residual based a posteriori error estimators for nonlinear problems in (Verfürth 13).
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Journals

T.M. Huang, W.W. Lin, V. Mehrmann
A NewtonType Method with Nonequivalence Deflation for Nonlinear Eigenvalue Problems Arising in Photonic Crystal Modeling.
SIAM J. Sci. Comput. 38(2):B191B218, 2016. DOI:10.1137/151004823 
S. Giani, L. Grubišić, A. Międlar, J. S. Ovall
Robust estimates for hpadaptive approximations of nonselfadjoint eigenvalue problems.
Numerische Mathematik, 2015. Published online, DOI:10.1007/s0021101507523
Proceedings

C. Conrads, V. Mehrmann, A. Międlar
Adaptive Numerical Solution of Eigenvalue Problems arising from Finite Element Models. AMLS vs. AFEM.
AMS Contemporary Mathematics, 658:197225, 2016. Published online, DOI: 10.1090/conm/658/13127