FUNDAMENTALNAYA
I PRIKLADNAYA MATEMATIKA

(FUNDAMENTAL AND APPLIED MATHEMATICS)

2011/2012, VOLUME 17, NUMBER 5, PAGES 103-127

**The exponential dichotomy on general approximation scheme**

V. Pastor

S. Piskarev

Abstract

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This paper is devoted to the numerical analysis of abstract parabolic
problems $u\text{'}(t)\; =\; Au(t)$, $u(0)\; =\; u0$ 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.

Location: http://mech.math.msu.su/~fpm/eng/k1112/k115/k11507h.htm

Last modified: October 18, 2012