Approximation of the spectrum of closed operators: the determination of normal modes of a rotating basin

Descloux, Jean; Luskin, Mitchell; Rappaz, Jacques

doi:10.1090/S0025-5718-1981-0595047-5

Approximation of the spectrum of closed operators: the determination of normal modes of a rotating basin
HTML articles powered by AMS MathViewer

by Jean Descloux, Mitchell Luskin and Jacques Rappaz PDF

Math. Comp. 36 (1981), 137-154 Request permission

Abstract:

This paper gives a theory of spectral approximation for closed operators in Banach spaces. The perturbation theory developed in this paper is applied to the study of a finite element procedure for approximating the spectral properties of a differential system modeling a fluid in a rotating basin.

References

J. H. Bramble and J. E. Osborn, Rate of convergence estimates for nonselfadjoint eigenvalue approximations, Math. Comp. 27 (1973), 525–549. MR 366029, DOI 10.1090/S0025-5718-1973-0366029-9
Philippe G. Ciarlet, The finite element method for elliptic problems, Studies in Mathematics and its Applications, Vol. 4, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978. MR 0520174
Jean Descloux, Error bounds for an isolated eigenvalue obtained by the Galerkin method, Z. Angew. Math. Phys. 30 (1979), no. 2, 167–176 (English, with German summary). MR 535977, DOI 10.1007/BF01601931
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 483400, DOI 10.1051/m2an/1978120200971
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 483400, DOI 10.1051/m2an/1978120200971
Tosio Kato, Perturbation theory for linear operators, Die Grundlehren der mathematischen Wissenschaften, Band 132, Springer-Verlag New York, Inc., New York, 1966. MR 0203473
Tosio Kato, Perturbation theory for nullity, deficiency and other quantities of linear operators, J. Analyse Math. 6 (1958), 261–322. MR 107819, DOI 10.1007/BF02790238
Horace Lamb, Hydrodynamics, 6th ed., Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1993. With a foreword by R. A. Caflisch [Russel E. Caflisch]. MR 1317348
P. D. Lax and R. S. Phillips, Local boundary conditions for dissipative symmetric linear differential operators, Comm. Pure Appl. Math. 13 (1960), 427–455. MR 118949, DOI 10.1002/cpa.3160130307
J.-L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications. Vol. 1, Travaux et Recherches Mathématiques, No. 17, Dunod, Paris, 1968 (French). MR 0247243
Mitchell Luskin, Convergence of a finite element method for the approximation of normal modes of the oceans, Math. Comp. 33 (1979), no. 146, 493–519. MR 521272, DOI 10.1090/S0025-5718-1979-0521272-6
Wendell H. Mills Jr., The resolvent stability condition for spectra convergence with application to the finite element approximation of noncompact operators, SIAM J. Numer. Anal. 16 (1979), no. 4, 695–703. MR 537281, DOI 10.1137/0716052
Wendell H. Mills Jr., Optimal error estimates for the finite element spectral approximation of noncompact operators, SIAM J. Numer. Anal. 16 (1979), no. 4, 704–718. MR 537282, DOI 10.1137/0716053
John E. Osborn, Spectral approximation for compact operators, Math. Comput. 29 (1975), 712–725. MR 0383117, DOI 10.1090/S0025-5718-1975-0383117-3

J. Phys. Oceanography

Frigyes Riesz and Béla Sz.-Nagy, Functional analysis, Frederick Ungar Publishing Co., New York, 1955. Translated by Leo F. Boron. MR 0071727
J. H. Wilkinson, The algebraic eigenvalue problem, Clarendon Press, Oxford, 1965. MR 0184422