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

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 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 element-Galerkin approximation of the eigenvalues and eigenvectors of selfadjoint problems*, Math. Comp.**52**(1989), no. 186, 275–297. MR**962210**, 10.1090/S0025-5718-1989-0962210-8**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****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****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**, 10.1090/S0025-5718-1981-0595047-5**6.**Erhard Heinz,*Beiträge zur Störungstheorie der Spektralzerlegung*, Math. Ann.**123**(1951), 415–438 (German). MR**0044747****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****8.**Tosio Kato,*Estimation of iterated matrices, with application to the von Neumann condition*, Numer. Math.**2**(1960), 22–29. MR**0109826****9.**Tosio Kato,*Perturbation theory for linear operators*, 2nd ed., Springer-Verlag, Berlin-New York, 1976. Grundlehren der Mathematischen Wissenschaften, Band 132. MR**0407617****10.**Andrew V. Knyazev,*New estimates for Ritz vectors*, Math. Comp.**66**(1997), no. 219, 985–995. MR**1415802**, 10.1090/S0025-5718-97-00855-7**11.**A. Pazy,*Semigroups of linear operators and applications to partial differential equations*, Applied Mathematical Sciences, vol. 44, Springer-Verlag, New York, 1983. MR**710486****12.**Marvin Rosenblum,*On the operator equation 𝐵𝑋-𝑋𝐴=𝑄*, Duke Math. J.**23**(1956), 263–269. MR**0079235****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**

Retrieve articles in *Mathematics of Computation of the American Mathematical Society*
with MSC (1991):
65N25,
65N30,
65F15

Retrieve articles in all journals with MSC (1991): 65N25, 65N30, 65F15

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/S0025-5718-00-01181-9

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