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An Optimal Adaptive FEM for Eigenvalue Clusters

Please always quote using this URN:urn:nbn:de:0296-matheon-12870
  • The analysis of adaptive finite element methods in practice immediately leads to eigenvalue clusters which requires the simultaneous marking in adaptive finite element methods. A first analysis for multiple eigenvalues of the recent work [Dai, He, Zhou, arXiv Preprint 1210.1846v2] introduces an adaptive method whose marking strategy is based on the element-wise sum of local error estimator contributions for multiple eigenvalues. This paper proves optimality of a practical adaptive algorithm for eigenvalue clusters for the eigenvalues of the Laplace operator in terms of nonlinear approximation classes. All estimates are explicit in the initial mesh-size, the eigenvalues and the cluster width to clarify the dependence of the involved constants.

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Metadaten
Author:Dietmar Gallistl
URN:urn:nbn:de:0296-matheon-12870
Referee:Volker Mehrmann
Document Type:Preprint, Research Center Matheon
Language:English
Date of first Publication:2014/05/14
Release Date:2014/05/14
Tag:eigenvalue problem, eigenvalue cluster, adaptive finite element method, optimality
Institute:Humboldt-Universität zu Berlin
MSC-Classification:65-XX NUMERICAL ANALYSIS / 65Nxx Partial differential equations, boundary value problems / 65N12 Stability and convergence of numerical methods
Preprint Number:1059
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