Abstract
This paper is devoted to the numerical analysis of abstract parabolic problem ; , with hyperbolic generator . 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.
Highlights
Many problems like approximation of attractors, traveling waves, shadowing e.c. involve the notion of dichotomy
In numerical analysis of such problems, it is very important to know if they keep some kind of exponential estimates uniformly in discretization parameter
The operator B ∈ B E has an exponential discrete dichotomy with data M, r, P if P ∈ B E is a projector in E and M, r are constants with 0 ≤ r < 1 such that the following properties hold i BkP P Bk for all k ∈ N, ii Bk I − P ≤ Mrk for all k ∈ N, iii B : B|R P : R P → R P is a homeomorphism that satisfies B−kP ≤ Mrk, k ∈ N
Summary
Many problems like approximation of attractors, traveling waves, shadowing e.c. involve the notion of dichotomy. We assume that the part σ of the spectrum of operator A f u∗ , which is located strictly to the right of the imaginary axis, consists of a finite number of eigenvalues with finite multiplicity. Since Fu∗ vtovt Eα for small v · , the estimates 1.6 are crucial to describe the behavior of solution of the problem 1.2 at the vicinity of the hyperbolic equilibrium point u∗. Coming back to solution of 1.4 for any two v0, vT ∈ UEα 0; ρ we consider the boundary value problem v t Au∗ v t Fu∗ v t , 0 ≤ t ≤ T, 1.7. In case one would like to discretize the problem 1.4 in space and time variables it is very important to know what will happen to estimates like 1.6 for approximation solutions. In this paper we are going to consider general approximation approach for keeping the dichotomy estimates 1.6 for approximations of trajectory u ·
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