\`x^2+y_1+z_12^34\`
Advanced Search
Article Contents
Article Contents

Shadowing for discrete approximations of abstract parabolic equations

Abstract Related Papers Cited by
  • 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]).
    Mathematics Subject Classification: Primary: 65J15, 65M12, 34G20; Secondary: 65N12, 65P40, 35K40.

    Citation:

    \begin{equation} \\ \end{equation}
  • 加载中
SHARE

Article Metrics

HTML views() PDF downloads(75) Cited by(0)

Access History

Other Articles By Authors

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return