Preprint des Projektes: SFB 701: Spektrale Strukturen und Topologische Methoden in der Mathematik - Projekt A2

Numerische Spektralanalyse unendlich-dimensionaler Transferoperatoren

07-014 Wolf-Jürgen Beyn, Sergey Piskarev.
Shadowing for discrete approximations of abstract parabolic equations

This paper is devoted to the numerical analysis of abstract semilinear parabolic problems

in some general Banach space E. We prove a shadowing Theorem that compares solutions of the continuous problem with those of a semidiscrete apprximation (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 diference methods. In this way we obtain a unified approach to shadowing results derived e.g. in the finite element context ([19, 20, 21]).