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A Newton-Type Method with Non-equivalence Deflation for Nonlinear Eigenvalue Problems Arising in Photonic Crystal Modeling

Please always quote using this URN:urn:nbn:de:0296-matheon-13344
  • The numerical simulation of the band structure of three-dimensional dispersive metallic photonic crystals with face-centered cubic lattices leads to large-scale nonlinear eigenvalue problems, which are very challenging due to a high dimensional subspace associated with the eigenvalue zero and the fact that the desired eigenvalues (with smallest real part) cluster near the zero eigenvalues. For the solution of the eigenvalue problem, a Newton-type iterative method is proposed and the nullspace-free method is applied to exclude the zero eigenvalues from the associated generalized eigenvalue problem. To find the successive eigenvalue/eigenvector pairs, we propose a new non-equivalence deflation method to transform converged eigenvalues to infinity, while all other eigenvalues remain unchanged. The deflated problem is then solved by the same Newton-type method, which uses a hybrid method that combines the Jacobi-Davidson, the shift-invert residual Arnoldi and nonlinear Arnoldi methods to compute the clustered eigenvalues. Numerical results illustrate that the method is robust even for the case of computing many eigenvalues in very large problems.

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Metadaten
Author:Tsung-Ming Huang, Wen-Wei Lin, Volker Mehrmann
URN:urn:nbn:de:0296-matheon-13344
Referee:Frank Schmidt
Document Type:Preprint, Research Center Matheon
Language:English
Date of first Publication:2015/02/02
Release Date:2015/02/02
Tag:Maxwell equation; dispersive metallic photonic crystals; non-equivalence deflation; nonlinear eigenvalue problem
Institute:Technische Universität Berlin
Project:D Optics and Electronics (Electronic and photonic devices) / D-OT3 Adaptive finite element methods for nonlinear parameter-dependent eigenvalue problems in photonic crystals
MSC-Classification:15-XX LINEAR AND MULTILINEAR ALGEBRA; MATRIX THEORY / 15Axx Basic linear algebra / 15A18 Eigenvalues, singular values, and eigenvectors
65-XX NUMERICAL ANALYSIS / 65Fxx Numerical linear algebra / 65F15 Eigenvalues, eigenvectors
Preprint Number:1076
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