Advertisement

On the Convergence of the Finite Difference Method for Nonlinear Ordinary Boundary Value Problems

  • Wolf-Jürgen Beyn
Chapter
Part of the International Series of Numerical Mathematics / Internationale Schriftenreihe zur Numerischen Mathematik / Serie Internationale D’Analyse Numerique book series (ISNM, volume 48)

Abstract

There are well known conditions which finite difference equations approximating a linear ordinary boundary value problem have to satisfy in order to guarantee consistency and stability of the method and hence convergence of the finite difference solutions. Furthermore, under analogous assumptions a local convergence theorem holds in the nonlinear case. In this paper we give two global versions of this local result, one which yields a global stability inequality for the finite difference equations and another one which shows that the number of solutions is the same for the difference equations as for the boundary value problem. Our results are illustrated by two examples.

Keywords

Local Result Linear Scheme Finite Difference Equation Root Condition Finite Difference Solution 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Atkinson, K.E.: The numerical evaluation of fixed points for completely continuous operators. SIAM J. Numer. Anal. (1973), 799-807.Google Scholar
  2. [2]
    Beyn, W.-J.: Zur Stabilität von Differenzenverfahren für Systeme linearer gewöhnlicher Randwertaufgaben. Numer. Math. 29 (1978), 209–226.CrossRefGoogle Scholar
  3. [3]
    Bohl, E.: On the numerical treatment of a class of discrete bifurcation problems. To appear in IAC, Istituto per le Applicazioni del Calcolo, “Mauro Picone”, Pubblicazioni Serie III.Google Scholar
  4. [4]
    Bohl, E., Lorenz, J.: Inverse monotonicity and difference schemes of higher order. A summary for two-point boundary value problems. To appear in Aequ. Math.Google Scholar
  5. [5]
    Collatz, L.: The numerical treatment of differential equations, 3rd ed. Berlin-Göttingen-Heidelberg, Springer 1966.Google Scholar
  6. [6]
    Crandall, M.G., Rabinowitz, P.H.: Bifurcation, perturbation of simple eigenvalues and linearized stability. Arch. Rat. Mech. Anal. 52 (1973), 161–180.CrossRefGoogle Scholar
  7. [7]
    Grigorieff, R.D.: Die Konvergenz des Rand-und Eigenwertproblems linearer gewöhnlicher Differentialgleichungen. Numer. Math. 15 (1970), 15–48.CrossRefGoogle Scholar
  8. [8]
    Keller, H.B.: Approximation methods for nonlinear problems with application to two point boundary value problems. Math. Comput. 29 (1975), 464–474.CrossRefGoogle Scholar
  9. [9]
    Kreiss, H.-O.: Difference approximations for boundary and eigenvalue problems for ordinary differential equations. Math. Comput. 26 (1972), 605–624.CrossRefGoogle Scholar
  10. [10]
    Meyer-Spasche, R.: Numerische Behandlung von elliptischen Randwertproblemen mit mehreren Lösungen und von MHD Gleichgewichtsproblemen. Max Planck Institut für Plasmaphysik, Garching bei München, 1975.Google Scholar
  11. [11]
    Protter, M.H., Weinberger, H.F.: Maximum principles in differential equations. Prentice Hall, Englewood Cliffs, N.J., 1967.Google Scholar
  12. [12]
    Shampine, L.F.: Boundary value problems for ordinary differential equations I. SIAM J. Numer. Anal. 5 (1968), 219–242.CrossRefGoogle Scholar
  13. [13]
    Stetter, H.J.: Analysis of discretization methods for ordinary differential equations. Springer Tracts in Natural Philosophy, vol. 23. Berlin-Heidelberg-New York, Springer, 1973.CrossRefGoogle Scholar
  14. [14]
    Stummel, F.: Stability and discrete convergence of differentiable mappings. Rev. Roum. Math. Pures et Appl. Tome XXI, No 1 (1976), 63–96.Google Scholar
  15. [15]
    Vainikko, G.: Funktionalanalysis der Diskretisierungs-methoden. Teubner Texte zur Mathematik, Leipzig, Teubner Verlag 1976.Google Scholar
  16. [16]
    Vainikko, G.: Approximate methods for nonlinear equations. Nonlinear Analysis 2, (1978), 647–687.CrossRefGoogle Scholar

Copyright information

© Springer Basel AG 1979

Authors and Affiliations

  • Wolf-Jürgen Beyn
    • 1
  1. 1.Institut für Numerische und instrumentelle MathematikUniversität MünsterMünsterGermany

Personalised recommendations