Galerkin eigenvector approximations
Author:
Christopher Beattie
Journal:
Math. Comp. 69 (2000), 14091434
MSC (1991):
Primary 65N25; Secondary 65N30, 65F15
Published electronically:
March 3, 2000
MathSciNet review:
1681128
Fulltext PDF Free Access
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Abstract: 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 subspaceand this occurs more rapidly than the underlying rate of convergence of the approximate eigenvectors. Both orthogonalGalerkin and PetrovGalerkin 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 of a pair of operators are developed as well.
 1.
Babuska, I., and J. Osborn (1991), Eigenvalue Problems, in ``Finite Element Methods'' Handbook of Numerical Analysis, Vol. 2, edited by P. G. Ciarlet and J. L. Lions. Elsevier Science Publisher (North Holland). CMP 91:14
 2.
I.
Babuška and J.
E. Osborn, Finite elementGalerkin approximation
of the eigenvalues and eigenvectors of selfadjoint problems, Math. Comp. 52 (1989), no. 186, 275–297. MR 962210
(89k:65132), http://dx.doi.org/10.1090/S00255718198909622108
 3.
Françoise
Chatelin, Spectral approximation of linear operators, Computer
Science and Applied Mathematics, Academic Press, Inc. [Harcourt Brace
Jovanovich, Publishers], New York, 1983. With a foreword by P. Henrici;
With solutions to exercises by Mario Ahués. MR 716134
(86d:65071)
 4.
Jean
Descloux, Nabil
Nassif, and Jacques
Rappaz, On spectral approximation. I. The problem of
convergence, RAIRO Anal. Numér. 12 (1978),
no. 2, 97–112, iii (English, with French summary). MR 0483400
(58 #3404a)
 5.
Jean
Descloux, Mitchell
Luskin, and Jacques
Rappaz, Approximation of the spectrum of
closed operators: the determination of normal modes of a rotating
basin, Math. Comp. 36
(1981), no. 153, 137–154. MR 595047
(83h:65123), http://dx.doi.org/10.1090/S00255718198105950475
 6.
Erhard
Heinz, Beiträge zur Störungstheorie der
Spektralzerlegung, Math. Ann. 123 (1951),
415–438 (German). MR 0044747
(13,471f)
 7.
Jan
Kadlec, The regularity of the solution of the Poisson problem in a
domain whose boundary is similar to that of a convex domain,
Czechoslovak Math. J. 14 (89) (1964), 386–393
(Russian, with English summary). MR 0170088
(30 #329)
 8.
Tosio
Kato, Estimation of iterated matrices, with application to the von
Neumann condition, Numer. Math. 2 (1960),
22–29. MR
0109826 (22 #711)
 9.
Tosio
Kato, Perturbation theory for linear operators, 2nd ed.,
SpringerVerlag, BerlinNew York, 1976. Grundlehren der Mathematischen
Wissenschaften, Band 132. MR 0407617
(53 #11389)
 10.
Andrew
V. Knyazev, New estimates for Ritz
vectors, Math. Comp. 66
(1997), no. 219, 985–995. MR 1415802
(97j:65090), http://dx.doi.org/10.1090/S0025571897008557
 11.
A.
Pazy, Semigroups of linear operators and applications to partial
differential equations, Applied Mathematical Sciences, vol. 44,
SpringerVerlag, New York, 1983. MR 710486
(85g:47061)
 12.
Marvin
Rosenblum, On the operator equation
𝐵𝑋𝑋𝐴=𝑄, Duke Math. J.
23 (1956), 263–269. MR 0079235
(18,54d)
 13.
J.
Wloka, Partial differential equations, Cambridge University
Press, Cambridge, 1987. Translated from the German by C. B. Thomas and M.
J. Thomas. MR
895589 (88d:35004)
 1.
 Babuska, I., and J. Osborn (1991), Eigenvalue Problems, in ``Finite Element Methods'' Handbook of Numerical Analysis, Vol. 2, edited by P. G. Ciarlet and J. L. Lions. Elsevier Science Publisher (North Holland). CMP 91:14
 2.
 Babuska, I., and J. Osborn (1989), Finite elementGalerkin approximation of the eigenvalues and eigenvectors of selfadjoint problems, Math. Comp. 52, pp. 275297. MR 89k:65132
 3.
 Chatelin, F. (1983), Spectral Approximation of Linear Operators, (Academic Press, New York)MR 86d:65071
 4.
 Descloux, J., N. Nassif, and J. Rappaz (1978), On spectral approximation, Part I: The problem of convergence, RAIRO Anal. Numér. 12, pp. 97112. MR 58:3404a
 5.
 Descloux, J., M. Luskin, and J. Rappaz (1981), Approximation of the spectrum of closed operators: the determination of normal modes of a rotating basin, Math. Comp. 36, pp. 137154. MR 83h:65123
 6.
 Heinz, E. (1951), Beiträge zur Störungstheorie der Spektralzerlegung, Math. Ann. 123, pp. 415438. MR 13:471f
 7.
 Kadlec, J. (1964), The regularity of the solution of the Poisson problem in a domain whose boundary is similar to that of a convex domain, Czech Math J. 14 (89), pp. 386393. (In Russian)MR 30:329
 8.
 Kato, T. (1960), Estimation of iterated matrices, with application to the von Neumann condition, Numerische Mathematik 2, pp. 2229.MR 22:711
 9.
 Kato, T. (1976), Perturbation Theory for Linear Operators, (Springer, Heidelberg) MR 53:11389
 10.
 Knyazev, A. (1997), New estimates for Ritz vectors, Math. Comp., 66 (219), pp. 985995. MR 97j:65090
 11.
 Pazy, A. (1983), Semigroups of Linear Operators and Applications to Partial Differential Equations, (Springer, Heidelberg) MR 85g:47061
 12.
 Rosenblum, M. (1956), On the operator equation , Duke Math. J. pp. 263269.MR 18:54d
 13.
 Wloka, J. (1987), Partial Differential Equations, (Cambridge Univ. Press, Cambridge)MR 88d:35004
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Additional Information
Christopher Beattie
Affiliation:
Department of Mathematics, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061 USA
Email:
beattie@math.vt.edu
DOI:
http://dx.doi.org/10.1090/S0025571800011819
PII:
S 00255718(00)011819
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 F496209610329
Article copyright:
© Copyright 2000
American Mathematical Society
