The exponential dichotomy on general approximation scheme

This paper is devoted to the numerical analysis of abstract parabolic problems $u'(t)=Au(t)$, $u(0)=u^0$ with hyperbolic generator $A$. We develop a general approach to establish a discrete dichotomy in a very general setting in the 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 such a way that the original initial value problem is reduced to initial value problems with exponentially decaying solutions in opposite time directions. 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 are naturally satisfied, in particular, for operators with compact resolvents and condensing semigroups and can be verified for the finite element method as well as finite difference methods.