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On estimators for eigenvalue/eigenvector approximations


Authors: Luka Grubisic and Jeffrey S. Ovall
Journal: Math. Comp. 78 (2009), 739-770
MSC (2000): Primary 65N25; Secondary 65N50, 65N15, 65N30
DOI: https://doi.org/10.1090/S0025-5718-08-02181-9
Published electronically: November 6, 2008
MathSciNet review: 2476558
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Abstract: We consider a large class of residuum based a posteriori eigenvalue/eigenvector estimates and present an abstract framework for proving their asymptotic exactness. Equivalence of the estimator and the error is also established. To demonstrate the strength of our abstract approach we present a detailed study of hierarchical error estimators for Laplace eigenvalue problems in planar polygonal regions. To this end we develop new error analysis for the Galerkin approximation which avoids the use of the strengthened Cauchy-Schwarz inequality and the saturation assumption, and gives reasonable and explicitly computable upper bounds on the discretization error. A brief discussion is also given concerning the design of estimators which are in the same spirit, but are based on different a posteriori techniques--notably, those of gradient recovery type.


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Additional Information

Luka Grubisic
Affiliation: Institut für reine und angewandte Mathematik, RWTH-Aachen, Templergraben 52, D-52062 Aachen, Germany
Address at time of publication: Department of Mathematics, Univ-Zagreb, Bijenicka 30, 10000 Zagreb, Croatia
Email: luka.grubisic@math.hr

Jeffrey S. Ovall
Affiliation: Max-Planck-Institut für Mathematik in den Naturwissenschaften, Inselstr. 22-26, D-04103 Leipzig, Germany
Address at time of publication: California Institute of Technology, Pasadena, California 91125-5000
Email: ovall@mis.mpg.de, jovall@acm.caltech.edu

DOI: https://doi.org/10.1090/S0025-5718-08-02181-9
Keywords: Eigenvalue problem, finite element method, a posteriori error estimates
Received by editor(s): February 20, 2007
Received by editor(s) in revised form: April 18, 2008
Published electronically: November 6, 2008
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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