Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 
 

 

Spectral approximation for compact operators


Author: John E. Osborn
Journal: Math. Comp. 29 (1975), 712-725
MSC: Primary 47A55; Secondary 65J05
DOI: https://doi.org/10.1090/S0025-5718-1975-0383117-3
MathSciNet review: 0383117
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • [1] P. M. ANSELONE, Collectively Compact Operator Approximation Theory, Prentice-Hall, Englewood Cliffs, N. J., 1971. MR 0443383 (56:1753)
  • [2] K. ATKINSON, "Convergence rates for approximate eigenvalues of compact integral operators," SIAM J. Numer. Anal., v. 12, 1975, pp. 213-222. MR 0438746 (55:11653)
  • [3] I. BABUŠKA, "The finite element method with Lagrangian multipliers," Numer. Math., v. 20, 1973, pp. 179-192. MR 0359352 (50:11806)
  • [4] I. BABUŠKA, Solution of Interface Problems by Homogenization. I, Technical Note BN-782, Institute for Fluid Dynamics and Applied Mathematics, University of Maryland, College Park, Md., 1974.
  • [5] I. BABUŠKA, Solution of Problems with Interfaces and Singularities, Proc. Sympos., Math. Res. Center, University of Wisconsin, Madison, Wis., April 1974.
  • [6] J. H. BRAMBLE & J. E. OSBORN, "Approximation of Steklov eigenvalues of nonselfadjoint second order elliptic operators," The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations (edited by A. K. Aziz), Academic Press, New York, 1972, pp. 387-408. MR 0431740 (55:4735)
  • [7] J. H. BRAMBLE & J. E. OSBORN, "Rate of convergence estimates for nonselfadjoint eigenvalue approximations," Math. Comp., v. 27, 1973, pp. 525-549. MR 0366029 (51:2280)
  • [8] J. H. BRAMBLE & A. H. SCHATZ, "Rayleigh-Ritz-Galerkin methods for Dirichlet's problem using subspaces without boundary conditions," Comm. Pure Appl. Math., v. 23, 1970, pp. 653-675. MR 42 #2690. MR 0267788 (42:2690)
  • [9] N. DUNFORD & J. T. SCHWARTZ, Linear Operators. II: Spectral Theory. Selfadjoint Operators in Hilbert Space, Interscience, New York, 1963. MR 32 #6181. MR 0188745 (32:6181)
  • [10] T. KATO, "Perturbation theory for nullity, deficiency and other quantities of linear operators," J. Analyse Math., v. 6, 1958, pp. 261-322. MR 21 #6541. MR 0107819 (21:6541)
  • [11] T. KATO, Perturbation Theory for Linear Operators, Die Grundlehren der Math. Wissenschaften, Band 132, Springer-Verlag, New York, 1966. MR 34 #3324. MR 0203473 (34:3324)
  • [12] H. KREISS, "Difference approximations for boundary and eigenvalue problems for ordinary differential equations," Math. Comp., v. 26, 1972, pp. 605-624. MR 0373296 (51:9496)
  • [13] D. C. LAY, Private communication.
  • [14] J.-L. LIONS & E. MAGENES, Problèmes aux limites non homogènes et applications. Vol. I, Travaux et Recherches Mathématiques, no. 17, Dunod, Paris, 1968. MR 40 #512. MR 0247243 (40:512)
  • [15] J. NITSCHE, "Ein Kriterium für die Quasi-Optimalität des Ritzschen Verfahrens," Numer. Math., v. 11, 1968, pp. 346-348. MR 38 #1823. MR 0233502 (38:1823)
  • [16] J. NITSCHE, "Über ein Variationsprinzip zur Lösung von Dirichlet-Problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unterworfen sind," Abh. Math. Sem. Univ. Hamburg, v. 36, 1970/71. MR 0341903 (49:6649)
  • [17] J. NITSCHE, "A projection method for Dirichlet-problems using subspaces with nearly zero boundary conditions." (Preprint.) MR 0426456 (54:14399)
  • [18] G. M. VAĬNIKKO, "Asymptotic error bounds for projection methods in the eigenvalue problem," Ž. Vyčisl. Mat. i Mat. Fiz., v. 4, 1964, pp. 405-425 = U. S. S. R. Comput. Math. and Math. Phys., v. 4, 1964, pp. 9-36. MR 31 #615. MR 0176340 (31:615)
  • [19] G. M. VAĬNIKKO, "On the rate of convergence of certain approximation methods of Galerkin type in an eigenvalue problem," Izv. Vysš. Učebn. Zaved. Matematika, v. 1966, no. 2(51), pp. 37-45; English transl., Amer. Math. Soc. Transl. (2), v. 86, 1970, pp. 249-259. MR 33 #6824; 41 #1462. MR 0198669 (33:6824)
  • [20] G. M. VAĬNIKKO, "Rapidity of convergence of approximation methods in the eigenvalue problem," Ž. Vyčisl. Mat. i Mat. Fiz., v. 7, 1967, pp. 977-987 = U. S. S. R. Comput. Math. and Math. Phys., v. 7, 1967, pp. 18-32. MR 36 #4798. MR 0221746 (36:4798)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 47A55, 65J05

Retrieve articles in all journals with MSC: 47A55, 65J05


Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1975-0383117-3
Keywords: Approximation of eigenvalues, approximation of generalized eigenvectors, nonselfadjoint operators, finite element method
Article copyright: © Copyright 1975 American Mathematical Society

American Mathematical Society