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July  2008, 10(1): 19-42. doi: 10.3934/dcdsb.2008.10.19

Shadowing for discrete approximations of abstract parabolic equations

Wolf-Jürgen Beyn 1, and  Sergey Piskarev 2,

1. 

Department of Mathematics, Bielefeld University, P.O. Box 100131, 33501 Bielefeld, Germany
2. 

Scientific Research Computer Center, Moscow State University, Vorobjevy Gory, Moscow 119899, Russian Federation

Received  February 2007 Revised  February 2008 Published  April 2008

This paper is devoted to the numerical analysis of abstract semilinear parabolic problems $u'(t) = Au(t) + f(u(t)), u(0)=u^0,$ in some general Banach space $E$. We prove a shadowing Theorem that compares solutions of the continuous problem with those of a semidiscrete approximation (time stays continuous) in the neighborhood of a hyperbolic equilibrium. We allow rather general discretization schemes following the theory of discrete approximations developed by F. Stummel, R.D. Grigorieff and G. Vainikko. We use a compactness principle to show that the decomposition of the flow into growing and decaying solutions persists for this general type of approximation. The main assumptions of our results are naturally satisfied for operators with compact resolvents and can be verified for finite element as well as finite difference methods. In this way we obtain a unified approach to shadowing results derived e.g. in the finite element context ([19, 20, 21]).
Keywords: hyperbolic equilibria, semidiscretization, abstract parabolic problems, Theory of shadowing, compact convergence of resolvents..
Mathematics Subject Classification: Primary: 65J15, 65M12, 34G20; Secondary: 65N12, 65P40, 35K4.
Citation: Wolf-Jürgen Beyn, Sergey Piskarev. Shadowing for discrete approximations of abstract parabolic equations. Discrete and Continuous Dynamical Systems - B, 2008, 10 (1) : 19-42. doi: 10.3934/dcdsb.2008.10.19
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