Advertisement

Abstract

There is a duality between the use of node functions and that of cell functions in finite difference methods. From the same extremum principle they give bounds in opposite directions for a domain functional. — A difference scheme is called coherent if the equations relative to successive mesh sizes don’t contradict each other. Such coherent schemes give the exact solutions in very elementary problems. In general they yield especially good approximations. — Not only in symmetric domains, also in the case of “generali zed symmetry” the number of unknowns can be reduced by splitting the problem. Not the domain itself, but the set of admissible functions in it has a certain symmetry.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Literaturverzeichnis

  1. [1]
    Chan, S.L.: On the possible application of group representations to some boundary value and eigenvalue problems of certain domains without classical symmetry. Zeitschr. Angew. Math. Phys. 27 (1976), 553–572 & 839-851.CrossRefGoogle Scholar
  2. [2]
    Collatz, L.: Konvergenz des Differenzverfahrens bei Eigenwertproblemen partieller Differentialgleichungen. Dtsch. Math. 3 (1938), 200–212.Google Scholar
  3. [3]
    Courant, R.: Variational methods for the solutions of problems of equilibrium and vibrations. Bull. Amer. Math. Soc. 49 (1943), 1–23.CrossRefGoogle Scholar
  4. [4]
    Hersch, J.: Récurrences d’ordre supérieur pour des équations aux différences. C.R. Acad. Sci. Paris 246 (1958), 364–367.Google Scholar
  5. [5]
    Hersch, J.: Contribution à la méthode des équations aux différences. Zeitschr. Angew. Math. Phys. 9a (1958), 129–180.CrossRefGoogle Scholar
  6. [6]
    Hersch, J.: Lower bounds for all eigenvalues by cell functions: a refined form of H.F. Weinberger’s method. Arch. Rat. Mech. Anal. 12 (1963), 361–366.CrossRefGoogle Scholar
  7. [7]
    Hersch, J.: Erweiterte Symmetrieeigenschaften von Lösungen gewisser linearer Rand-und Eigenwertprobleme. J. reine angew. Math. 218 (1965), 143–158.Google Scholar
  8. [8]
    Hersch, J.: Sur les fonctions propres des membranes vibrantes couvrant un secteur symétrique de polygone régulier ou de domaine périodique. Comment. Math. Helv. 41 (1966-67), 222–236.CrossRefGoogle Scholar
  9. [9]
    Hersch, J.: Eine Kohärenzforderung für Differenzengleichungen. ISNM 19 (1974), 121–124.Google Scholar
  10. [10]
    Pólya, G.: Sur une interprétation de la méthode des différences finies qui peut fournir des bornes supérieures ou inférieures. C.R. Acad. Sci. Paris 235 (1952), 995–997.Google Scholar
  11. [11]
    Velte, W.: Eigenwertschranken mit finiten Differenzen beim Stokesschen Eigenwertproblem. ISNM 43 (1979), 176–188.Google Scholar
  12. [12]
    Weinberger, H.F.: Lower bounds for higher eigenvalues by finite difference methods. Pacific J. Math. 8 (1958), 339–368.CrossRefGoogle Scholar

Copyright information

© Springer Basel AG 1983

Authors and Affiliations

  • Joseph Hersch
    • 1
  1. 1.MathematikETH-ZentrumZürichSchweiz

Personalised recommendations