Spectral approximation for compact operators

Osborn, John E.

doi:10.1090/S0025-5718-1975-0383117-3

Spectral approximation for compact operators
HTML articles powered by AMS MathViewer

by John E. Osborn PDF

Math. Comp. 29 (1975), 712-725 Request permission

Abstract:

In this paper a general spectral approximation theory is developed for compact operators on a Banach space. Results are obtained on the approximation of eigenvalues and generalized eigenvectors. These results are applied in a variety of situations.

References

Philip M. Anselone, Collectively compact operator approximation theory and applications to integral equations, Prentice-Hall Series in Automatic Computation, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1971. With an appendix by Joel Davis. MR 0443383
Kendall Atkinson, Convergence rates for approximate eigenvalues of compact integral operators, SIAM J. Numer. Anal. 12 (1975), 213–222. MR 438746, DOI 10.1137/0712020
Ivo Babuška, The finite element method with Lagrangian multipliers, Numer. Math. 20 (1972/73), 179–192. MR 359352, DOI 10.1007/BF01436561

Solution of Interface Problems by Homogenization

Solution of Problems with Interfaces and Singularities

J. H. Bramble and J. E. Osborn, Approximation of Steklov eigenvalues of non-selfadjoint second order elliptic operators, The mathematical foundations of the finite element method with applications to partial differential equations (Proc. Sympos., Univ. Maryland, Baltimore, Md., 1972) Academic Press, New York, 1972, pp. 387–408. MR 0431740
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
James H. Bramble and Alfred H. Schatz, Rayleigh-Ritz-Galerkin methods for Dirichlet’s problem using subspaces without boundary conditions, Comm. Pure Appl. Math. 23 (1970), 653–675. MR 267788, DOI 10.1002/cpa.3160230408
Nelson Dunford and Jacob T. Schwartz, Linear operators. Part II: Spectral theory. Self adjoint operators in Hilbert space, Interscience Publishers John Wiley & Sons, New York-London, 1963. With the assistance of William G. Bade and Robert G. Bartle. MR 0188745
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
Tosio Kato, Perturbation theory for linear operators, Die Grundlehren der mathematischen Wissenschaften, Band 132, Springer-Verlag New York, Inc., New York, 1966. MR 0203473
Heinz-Otto Kreiss, Difference approximations for boundary and eigenvalue problems for ordinary differential equations, Math. Comp. 26 (1972), 605–624. MR 373296, DOI 10.1090/S0025-5718-1972-0373296-3
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
J. Nitsche, Ein Kriterium für die Quasi-Optimalität des Ritzschen Verfahrens, Numer. Math. 11 (1968), 346–348 (German). MR 233502, DOI 10.1007/BF02166687
J. Nitsche, Über ein Variationsprinzip zur Lösung von Dirichlet-Problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unterworfen sind, Abh. Math. Sem. Univ. Hamburg 36 (1971), 9–15 (German). MR 341903, DOI 10.1007/BF02995904
J. Nitsche, On Dirichlet problems using subspaces with nearly zero boundary conditions, The mathematical foundations of the finite element method with applications to partial differential equations (Proc. Sympos., Univ. Maryland, Baltimore, Md., 1972) Academic Press, New York, 1972, pp. 603–627. MR 0426456
G. M. Vainikko, Asymptotic error bounds for projection methods in the eigenvalue problem, Ž. Vyčisl. Mat i Mat. Fiz. 4 (1964), 405–425 (Russian). MR 176340
G. M. Vainikko, On the rate of convergence of certain approximation methods of Galerkin type in eigenvalue problems, Izv. Vysš. Učebn. Zaved. Matematika 1966 (1966), no. 2 (51), 37–45 (Russian). MR 0198669
G. M. Vainikko, Rapidity of convergence of approximation methods in eigenvalue problems, Ž. Vyčisl. Mat i Mat. Fiz. 7 (1967), 977–987 (Russian). MR 221746