On Estimators for Eigenvalue/Eigenvector Approximations

Luka Grubisic, and Jeffrey Ovall
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Submission date: 20. Feb. 2007
Pages: 31
published in: Mathematics of computation, 78 (2009) 266, p. 739-770 
DOI number (of the published article): 10.1090/S0025-5718-08-02181-9
MSC-Numbers: 65N15, 65N25, 65N30
Keywords and phrases: eigenvalue problems, finite element methods, a posteriori error estimation
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Abstract:
We consider a large class of residuum based \textit{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 a use of the strengthened Cauchy-Schwarz inequality and the saturation assumption, and gives reasonable and explicitly computable upper bounds on the discretization error. Brief discussion is also given concerning the design of estimators which are in the same spirit, but are based on different \textit{a posteriori} techniques -- notably, those of gradient recovery type.