The Exponential Dichotomy under Discretization on General Approximation Scheme

This paper is devoted to the numerical analysis of abstract parabolic problem ()=();(0)=0, with hyperbolic generator . We are developing a general approach to establisha discrete dichotomy in a very general setting in case of discrete approximation in space andtime. It is a well-known fact that the phase space in the neighborhood of thehyperbolic equilibrium can be split in a such way that the original initial value problem is reducedto initial value problems with exponential decaying solutions in opposite time direction. We usethe theory of compact approximation principle and collectively condensing approximation toshow 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 withcompact resolvents and condensing semigroups and can be verified for finite element as well asfinite difference methods