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On a numerical Liapunov-Schmidt method for operator equations

Ein numerisches Liapunov-Schmidt-Verfahren für Operatorgleichungen

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Abstract

Let, for a higher singular pointx 0 of an operator equationG(x 0)=0 and the kernels of the respective derivativesG′(x 0) andG′(x 0)*, see [1], approximations be available. We present a method to numerically compute the manifolds bifurcating atx 0. 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.

Zusammenfassung

Für einen höheren singulären Punktx 0 einer OperatorgleichungG(x 0)=0 und die Nullräume der jeweiligen AbleitungenG′(x 0) undG′(x 0)*, siehe [1], seien Approximatioinen bekannt. Dann definieren wir ein numerisches Verfahren zur Berechnung der inx 0 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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  1. Fachbereich Mathematik, Philipps-Universität Marburg, D-35032, Marburg/Lahn, Federal Republic of Germany

    K. Böhmer

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  1. K. Böhmer
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Dedicated to Prof. Dr. Hans Stetter on the occasion of his 63rd birthday.

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Böhmer, K. On a numerical Liapunov-Schmidt method for operator equations. Computing 51, 237–269 (1993). https://doi.org/10.1007/BF02238535

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