Summary
We study evolution equations of typeu t =Au+Nu with polynomial nonlinearityN, in a Banach spaceB which lies in a Hilbert space. Under the restriction that the nonlinear operatorN is “finitely reproducing relative to the orthonormal sequencee i generated byAu=λu, the explicitly known Faedo-Galerkin approximations of the evolution equation can be estimated. The “reproducing” property is shown in the special case of a diffusion equation with Neumann boundary conditions and a nonlinearity of third degree. We study the numerical behavior of the approximations.
Zusammenfassung
Wir behandeln gewisse Evolutionsgleichungen der Artu t =Au+Nu, mit polynomialer NichtlinearitätN, in einem BanachraumB der in einem Hilbertraum liegt. Unter der Voraussetzung, daß der OperatorN, endlich reproduzierend bezüglich einer Orthonormalfolgee i ist, die durchAu=λu erzeugt wird, können die explizit bekannten Faedo-Galerkin-Approximationen der Evolutionsgleichung berechnet werden. Für den Spezialfall einer Diffusionsgleichung mit einer polynomialen Nichtlinearität dritten Grades und Neumann-Randbedingungen, wird die „reproduzierende Eigenschaft“ bewiesen und das numerische Verhalten der Approximationen untersucht.
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Supported in part by the DAAD at the University of Cologne, West Germany.
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Fiebig, M. Approximation of evolution equations with polynomial reproducing nonlinearities. Z. angew. Math. Phys. 37, 230–243 (1986). https://doi.org/10.1007/BF00945084
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DOI: https://doi.org/10.1007/BF00945084