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MiS Preprint
18/2007

On Estimators for Eigenvalue/Eigenvector Approximations

Luka Grubišić and Jeffrey Ovall

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 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 a posteriori techniques -- notably, those of gradient recovery type.

Received:
Feb 20, 2007
Published:
Feb 20, 2007
MSC Codes:
65N15, 65N25, 65N30
Keywords:
eigenvalue problems, finite element methods, a posteriori error estimation

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inJournal
2009 Repository Open Access
Luka Grubisic and Jeffrey S. Ovall

On estimators for eigenvalue/eigenvector approximations

In: Mathematics of computation, 78 (2009) 266, pp. 739-770