Bounds for Eigenvalues with the Use of Finite Elements

  • Henning Behnke
  • Ulrich Mertins


Verified upper and lower bounds for the smallest eigenvalues of eigenvalue problems with self-adjoint partial differential equations are computed. Upper bounds are obtained by the Rayleigh-Ritz method, a suitable Goerisch method provides lower bounds. The trial functions are constructed with the use of finite elements. All computations are carried out with intervaI arithmetic thus the results are protected against rounding errors. Numerical results are given for the L-shaped membrane.


Eigenvalue Problem Small Eigenvalue Trial Function Interval Arithmetic Matrix Eigenvalue Problem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Behnke H. (1991) The calculation of guaranteed bounds for eigenvalues using complementary variational principles. Computing 47, 11–27MATHGoogle Scholar
  2. 2.
    Behnke H., and Goerisch F. (1994) Inclusions for eigenvalues of selfadjoint problems. In: Herzberger, J. (Ed.) Topics in validated computations. Elsevier, Amsterdam, Lausanne, New York, Oxford, Shannon, Tokyo, 277–322Google Scholar
  3. 3.
    Behnke H., Mertins U., Plum M., and Wieners C. (2000) Eigenvalue inclusions via domain decomposition. Proc. R. Soc. Lond. A 456, 2717–2730CrossRefGoogle Scholar
  4. 4.
    Fox L., Henrici P., and Moler C. (1967) Approximations and bounds for eigenvalues of elliptic operators. SIAM J. Numer. Anal. 4, 89–102MATHCrossRefGoogle Scholar
  5. 5.
    Goerisch F. (1980) Eine Verallgemeinerung eines Verfahrens von N.J. Lehmann zur Einschließung von Eigenwerten. Wiss. Z. Tech. Univ. Dresden 29, 429–431Google Scholar
  6. 6.
    Goerisch F., and Haunhorst H. (1985) Eigenwertschranken für Eigenwertaufgaben mit partiellen Differentialgleichungen. Z. Angew. Math. Mech. 65, 129–135MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Knüppel O. (1994) Profil / bias- a fast intervallibrary. Computing 53, 277–287MATHGoogle Scholar
  8. 8.
    Lehmann N.J. (1949) and (1950) Beiträge zur Lösung linearer Eigenwertproblerne I und 11. Z. Angew. Math. Mech. 29,341-356 and 30,.1-16Google Scholar
  9. 9.
    Maehly H.J. (1952) Ein neues Variationsverfahren zur genäherten Berechnung der Eigenwerte hermitescher Operatoren. Helv. Phys. Acta 25, 547–568Google Scholar
  10. 10.
    Mertins U. (1991) Zur Konvergenz des Rayleigh-Ritz-Verfahrens bei Eigenwertaufgaben. Numer. Math. 59, 667–682MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Mertins U. (1992) Asymptotische Fehlerschranken für Rayleigh-Ritz-Approximationen selbstadjungierter Eigenwertaufgaben. Numer. Math. 63, 227–241MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Mertins U. (1996) On the convergence of the Goerisch method for self-adjoint eigenvalue problems with arbitrary spectrum. Z. Anal. Anwendungen 15, 661–686MathSciNetMATHGoogle Scholar
  13. 13.
    Klatte R., Kulisch U., Law C., and Rauch M. (1993) C-XSC- A C++ Class Library for Extended Scientific Computing. Springer, Heidelberg, New YorkGoogle Scholar
  14. 14.
    Klatte R., Kulisch U., Neaga M., Ratz D., and Ullrich Ch. (1991) Pascal-XSC - Language Reference with Examples. Springe, Heidelberg, New YorkGoogle Scholar
  15. 15.
    Schwarz H.-R. (1980) Methode der finiten Elemente. Teubner, StuttgartGoogle Scholar
  16. 16.
    Schwarz H.-R. (1988) Finite element methods. Academic Press Inc., LondonMATHGoogle Scholar
  17. 17.
    Weinberger H.F. (1974) Variational Methods for Eigenvalue Approximation. Regional Conference Series in Applied Mathematics, 15. SIAM, PiladelphiaGoogle Scholar
  18. 18.
    Zimmermann S., and Mertins U. (1995) Variational bounds to eigenvalues of self-adjoint problems with arbitrary spectrum. Z. Anal. Anwend. 14, 327–345MATHGoogle Scholar

Copyright information

© Springer-Verlag Wien 2001

Authors and Affiliations

  • Henning Behnke
    • 1
  • Ulrich Mertins
    • 1
  1. 1.TU ClausthalInstitut für MathematikClausthal-ZellerfeldGermany

Personalised recommendations