Computing

, Volume 51, Issue 3–4, pp 237–269

On a numerical Liapunov-Schmidt method for operator equations

  • K. Böhmer
Article

Abstract

Let, for a higher singular pointx0 of an operator equationG(x0)=0 and the kernels of the respective derivativesG′(x0) andG′(x0)*, see [1], approximations be available. We present a method to numerically compute the manifolds bifurcating atx0. In particular, the question of convergence of the numerical to the exact solution is studied by proving stability and convergence for solution parts of different order of magnitude. Different approaches are presented and applied to elliptic problems.

AMS Subject Classification

65J05 65J15 65N10 65N99 

Key words

Higher singular points numerical Liapunov-Schmidt method bifurcating manifolds nonlinear elliptic problems 

Ein numerisches Liapunov-Schmidt-Verfahren für Operatorgleichungen

Zusammenfassung

Für einen höheren singulären Punktx0 einer OperatorgleichungG(x0)=0 und die Nullräume der jeweiligen AbleitungenG′(x0) undG′(x0)*, siehe [1], seien Approximatioinen bekannt. Dann definieren wir ein numerisches Verfahren zur Berechnung der inx0 abzweigenden Lösungsmannigfaltigkeiten. Die Frage der Konvergenz der numerischen gegen die exakte Lösung wird studiert durch Nachweis der entsprechenden Stabilitäts- und Konvergenzeigenschaften von Lösungsanteilen verschiedener Größenordnungen. Das Verfahren wird angewandt auf ein elliptisches Problem.

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Copyright information

© Springer-Verlag 1993

Authors and Affiliations

  • K. Böhmer
    • 1
  1. 1.Fachbereich MathematikPhilipps-Universität MarburgMarburg/LahnFederal Republic of Germany

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