Numerische Mathematik

, Volume 29, Issue 2, pp 159–171

Approximation in variationally posed eigenvalue problems

  • William G. Kolata
Article

Summary

In this paper bounds are established for the error in Ritz-Galerkin approximation of variationally posed eigenvalue problems. For the most part these bounds are upper bounds, but lower bounds for the error are established for certain selfadjoint problems.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Babuška, I.: Survey lectures on the mathematical foundations of the finite element method. The mathematical foundations of the finite element method with applications to partial differential equations (A.K. Aziz, ed.), pp. 5–359. New York: Academic Press 1973Google Scholar
  2. 2.
    Babuška, I., Osborn, J.E.: Numerical treatment of eigenvalue problems for equations with discontinuous coefficients. Tech. Note BN-853, University of Maryland, USA, April 1977Google Scholar
  3. 3.
    Bramble, J.H., Osborn, J.E.: Rate of convergence estimates for nonselfadjoint eigenvalue approximation. Math. Comput.27, 525–549 (1973)Google Scholar
  4. 4.
    Chatelin, F.: La méthode de Galerkin. Ordre de convergence des éléments propres. C.R. Acad. Sci. Paris Sér. A278, 1213–1215 (1974)Google Scholar
  5. 5.
    Chatelin, F., Lemordant, J.: Error bounds in the approximation of eigenvalues of differential and integral operators. J. Math. Anal. Appl. (to appear)Google Scholar
  6. 6.
    Dunford, N., Schwartz, J.T.: Linear operators. II. Spectral theory, selfadjoint operators in Hilbert space. New York: Interscience 1963Google Scholar
  7. 7.
    Fix, G.M.: Eigenvalue approximation by the finite element method. Advances in Math.10, 300–316 (1973)Google Scholar
  8. 8.
    Grigorieff, R.D.: Diskrete Approximation von Eigenvertproblemen. I. Qualitative Konvergenz. Numer. Math.24, 355–374 (1975)—II. Konvergenzordnung. ibid, 415–433 (1975)—III. Asymptotische Entwicklungen. ibid25, 79–97 (1975)Google Scholar
  9. 9.
    Kato, T.: Perturbation theory for linear operators. Berlin-Heidelberg-New York: Springer 1966Google Scholar
  10. 10.
    Nemat-Nasser, S.: General variational methods for elastic waves in composites. J. Elasticity2, 73–90 (1972)Google Scholar
  11. 11.
    Osborn, J.E.: Spectral approximation for compact operators. Math. Comput.29, 712–725 (1975)Google Scholar
  12. 12.
    Taylor, A.E.: Introduction to functional analysis. New York: Wiley 1958Google Scholar
  13. 13.
    Weinberger, H.F.: Variational methods for eigenvalue approximation. Philadelphia: Society for Industrial and Applied Mathematics 1974Google Scholar

Copyright information

© Springer-Verlag 1978

Authors and Affiliations

  • William G. Kolata
    • 1
  1. 1.Department of Mathematics, Statistics and Computer ScienceThe American UniversityWashington, DCUSA

Personalised recommendations