An optimal adaptive FEM for eigenvalue clusters

Dietmar Gallistl

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19 Citations (Scopus)


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 et al. (arXiv Preprint 1210.1846v2, 2013) 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 the optimality of a practical adaptive algorithm based on a lowest-order conforming finite element method 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.
Original languageEnglish
Pages (from-to)467-496
Number of pages30
JournalNumerische Mathematik
Issue number3
Publication statusPublished - 2015
Externally publishedYes

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