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Adaptive Numerical Solution of Eigenvalue Problems arising from Finite Element Models. AMLS vs. AFEM

Please always quote using this URN:urn:nbn:de:0296-matheon-13364
  • We discuss adaptive numerical methods for the solution of eigenvalue problems arising either from the finite element discretization of a partial differential equation (PDE) or from discrete finite element modeling. When a model is described by a partial differential equation, the adaptive finite element method starts from a coarse finite element mesh which, based on a posteriori error estimators, is adaptively refined to obtain eigenvalue/eigenfunction approximations of prescribed accuracy. This method is well established for classes of elliptic PDEs, but is still in its infancy for more complicated PDE models. For complex technical systems, the typical approach is to directly derive finite element models that are discrete in space and are combined with macroscopic models to describe certain phenomena like damping or friction. In this case one typically starts with a fine uniform mesh and computes eigenvalues and eigenfunctions using projection methods from numerical linear algebra that are often combined with the algebraic multilevel substructuring to achieve an adequate performance. These methods work well in practice but their convergence and error analysis is rather difficult. We analyze the relationship between these two extreme approaches. Both approaches have their pros and cons which are discussed in detail. Our observations are demonstrated with several numerical examples.

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Metadaten
Author:Christoph Conrads, Volker Mehrmann, Agnieszka Miedlar
URN:urn:nbn:de:0296-matheon-13364
Referee:Caren Tischendorf
Document Type:Preprint, Research Center Matheon
Language:English
Date of first Publication:2015/02/27
Release Date:2015/02/27
Tag:AFEM; AMLS; eigenvalue problems
Institute:Research Center Matheon
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:65-XX NUMERICAL ANALYSIS / 65Fxx Numerical linear algebra / 65F15 Eigenvalues, eigenvectors
65-XX NUMERICAL ANALYSIS / 65Nxx Partial differential equations, boundary value problems / 65N25 Eigenvalue problems
65-XX NUMERICAL ANALYSIS / 65Nxx Partial differential equations, boundary value problems / 65N30 Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods
Preprint Number:1079
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