Shadowing for discrete approximations of abstract parabolic equations

Beyn W-J, Piskarev S (2008)
Discrete and Continuous Dynamical Systems-Series B 10(1): 19-42.

Zeitschriftenaufsatz | Veröffentlicht | Englisch
 
Download
Es wurde kein Volltext hochgeladen. Nur Publikationsnachweis!
Autor*in
Beyn, Wolf-JürgenUniBi; Piskarev, Sergey
Abstract / Bemerkung
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 ([ , , ]).
Stichworte
theory of shadowing; hyperbolic; equilibria; semidiscretization; compact convergence of resolvents; abstract parabolic problems
Erscheinungsjahr
2008
Zeitschriftentitel
Discrete and Continuous Dynamical Systems-Series B
Band
10
Ausgabe
1
Seite(n)
19-42
ISSN
1531-3492
Page URI
https://pub.uni-bielefeld.de/record/1587740

Zitieren

Beyn W-J, Piskarev S. Shadowing for discrete approximations of abstract parabolic equations. Discrete and Continuous Dynamical Systems-Series B. 2008;10(1):19-42.
Beyn, W. - J., & Piskarev, S. (2008). Shadowing for discrete approximations of abstract parabolic equations. Discrete and Continuous Dynamical Systems-Series B, 10(1), 19-42. doi:10.3934/dcdsb.2008.10.19
Beyn, W. - J., and Piskarev, S. (2008). Shadowing for discrete approximations of abstract parabolic equations. Discrete and Continuous Dynamical Systems-Series B 10, 19-42.
Beyn, W.-J., & Piskarev, S., 2008. Shadowing for discrete approximations of abstract parabolic equations. Discrete and Continuous Dynamical Systems-Series B, 10(1), p 19-42.
W.-J. Beyn and S. Piskarev, “Shadowing for discrete approximations of abstract parabolic equations”, Discrete and Continuous Dynamical Systems-Series B, vol. 10, 2008, pp. 19-42.
Beyn, W.-J., Piskarev, S.: Shadowing for discrete approximations of abstract parabolic equations. Discrete and Continuous Dynamical Systems-Series B. 10, 19-42 (2008).
Beyn, Wolf-Jürgen, and Piskarev, Sergey. “Shadowing for discrete approximations of abstract parabolic equations”. Discrete and Continuous Dynamical Systems-Series B 10.1 (2008): 19-42.