Advances in Numerical Analysis
Volume 2011 (2011), Article ID 582740, 22 pages
Research Article

The Exponential Dichotomy under Discretization on General Approximation Scheme

1Department de Mathemàtica Aplicada, University de València, 46100 Burjassot, Valencia, Spain
2Scientific Research Computer Center, Moscow State University, Vorobjevy Gory, Moscow 119899, Russia

Received 31 August 2011; Accepted 15 November 2011

Academic Editor: William J. Layton

Copyright © 2011 Javier Pastor and Sergey Piskarev. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


This paper is devoted to the numerical analysis of abstract parabolic problem 𝑢 ( 𝑡 ) = 𝐴 𝑢 ( 𝑡 ) ; 𝑢 ( 0 ) = 𝑢 0 , with hyperbolic generator 𝐴 . We are developing a general approach to establish a discrete dichotomy in a very general setting in case of discrete approximation in space and time. It is a well-known fact 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 decaying solutions in opposite time direction. We use the theory of compact approximation principle and collectively condensing approximation to show that such a decomposition of the flow persists under rather general approximation schemes. The main assumption of our results is naturally satisfied, in particular, for operators with compact resolvents and condensing semigroups and can be verified for finite element as well as finite difference methods.