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Approximations of Parabolic Equations at the Vicinity of Hyperbolic Equilibrium Point

2014; Taylor & Francis; Volume: 35; Issue: 10 Linguagem: Inglês

10.1080/01630563.2014.884580

1532-2467

Qingjie Cao, Javier Pastor, Sergey Piskarev, Stefan Siegmund,

Advanced Numerical Methods in Computational Mathematics

This article is devoted to the numerical analysis of the abstract semilinear parabolic problem u′(t) = Au(t) + f(u(t)), u(0) = u 0, in a Banach space E. We are developing a general approach to establish a discrete dichotomy in a very general setting and prove shadowing theorems that compare solutions of the continuous problem with those of discrete approximations in space and time. In [3 W.-J. Beyn and S. Piskarev ( 2008 ). Shadowing for discrete approximations of abstract parabolic equations . Discrete Contin. Dyn. Syst., B 10 : 19 – 42 .[Crossref], [Web of Science ®] , [Google Scholar]] the discretization in space was constructed under the assumption of compactness of the resolvent. It is a well-known fact (see [10 S. Larsson ( 1999 ). Numerical analysis of semilinear parabolic problems . In: The Graduate Student's Guide to Numerical Analysis ’98 ( M. Ainsworth , ed.). Lecture Notes from the 8th EPSRC Summer School in Numerical Analysis . Leicester , UK , July 5–17. 1998. Ser. Comput. Math. 26:83–117; Springer, Berlin .[Crossref] , [Google Scholar], 11 S. Larsson and J. M. Sanz-Serna (1994). The behavior of finite element solutions of semilinear parabolic problems near stationary points. SIAM J. Numer. Anal. 31:1000–1018.[Crossref], [Web of Science ®] , [Google Scholar]]) that the phase space in the neighborhood of the hyperbolic equilibrium can be split in a such way that the original initial value problem is reduced to initial value problems with exponential bounded solutions on the corresponding subspaces. We show that such a decomposition of the flow persists under rather general approximation schemes, utilizing a uniform condensing property. The main assumption of our results are naturally satisfied, in particular, for operators with compact resolvents and condensing semigroups and can be verified for finite elements as well as finite differences methods.