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Galerkin eigenvector approximations

Author: Christopher Beattie
Journal: Math. Comp. 69 (2000), 1409-1434
MSC (1991): Primary 65N25; Secondary 65N30, 65F15
Published electronically: March 3, 2000
MathSciNet review: 1681128
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How close are Galerkin eigenvectors to the best approximation available out of the trial subspace? Under a variety of conditions the Galerkin method gives an approximate eigenvector that approaches asymptotically the projection of the exact eigenvector onto the trial subspace--and this occurs more rapidly than the underlying rate of convergence of the approximate eigenvectors. Both orthogonal-Galerkin and Petrov-Galerkin methods are considered here with a special emphasis on nonselfadjoint problems, thus extending earlier studies by Chatelin, Babuska and Osborn, and Knyazev. Consequences for the numerical treatment of elliptic PDEs discretized either with finite element methods or with spectral methods are discussed. New lower bounds to the $sep$ of a pair of operators are developed as well.

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Additional Information

Christopher Beattie
Affiliation: Department of Mathematics, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061 USA

Keywords: Galerkin, eigenvector asymptotics, finite elements, spectral methods, \textit{sep}.
Received by editor(s): January 6, 1998
Received by editor(s) in revised form: July 10, 1998, and October 9, 1998
Published electronically: March 3, 2000
Additional Notes: This work was supported under the auspices of AFOSR Grant F49620-96-1-0329
Article copyright: © Copyright 2000 American Mathematical Society