

Preprint 42/2010
A framework for robust eigenvalue and eigenvector error estimation and Ritz value convergence enhancement
Luka Grubišić, Jeffrey Ovall, and Randolph Bank
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Submission date: 05. Aug. 2010
Pages: 32
published in: Applied numerical mathematics, 66 (2013), p. 1-29
DOI number (of the published article): 10.1016/j.apnum.2012.11.004
Bibtex
MSC-Numbers: 65N30, 65N25, 65N15
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Abstract:
We present a general framework for the
a posteriori estimation and enhancement of error in
eigenvalue/eigenvector computations for symmetric and elliptic
eigenvalue problems, and provide detailed analysis of a specific
and important example within this framework--finite element methods
with continuous, affine elements. A distinguishing feature of the
proposed approach is that it provides provably efficient and reliable
error estimation under very realistic assumptions, not only for
single, simple eigenvalues, but also for clusters which may contain
degenerate eigenvalues. We reduce the study of the
eigenvalue/eigenvector error estimators to the study of associated
boundary value problems, and make use of the wealth of knowledge
available for such problems. Our choice of a posteriori
error estimator, computed using hierarchical bases, very naturally
offers a means not only for estimating error in eigenvalue/eigenvector
computations, but also cheaply accelerating the convergence of these
computations--sometimes with convergence rates which are nearly twice that
of the unaccelerated approximations.