Abstract
This paper discusses two numerical schemes that can be used to approximate inertial manifolds whose existence is given by one of the standard methods of proof. The methods considered are fully numerical, in that they take into account the need to interpolate the approximations of the manifold between a set of discrete gridpoints. As all the discretisations are refined the approximations are shown to converge to the true manifold.
Highlights
Thus hIn what follows P = Pn and Q = Qn. In the continuous case the method involves finding the fixed point of an integral operator, and this corresponds to an invariant manifold for the equation (Chow et al [1]; Foias et al [9]; Henry [13]; Miklavcic [19]; Rodriguez-Bernal [28]; Temam [30])
Introduction & summarySince their introduction by Foias et al [12], inertial manifolds have been an active area of research
In this paper we develop two computational methods for calculating to within an arbitrary degree of accuracy any inertial manifold whose existence can be proved when a standard spectral gap condition holds
Summary
In what follows P = Pn and Q = Qn. In the continuous case the method involves finding the fixed point of an integral operator, and this corresponds to an invariant manifold for the equation (Chow et al [1]; Foias et al [9]; Henry [13]; Miklavcic [19]; Rodriguez-Bernal [28]; Temam [30]). For a given Lipschitz function φ ∈ F n and point p0 ∈ Rn denote by p(t) the solution of the equation dp/dt + Ap + P f (p + φ(p)) = 0 p(0) = p0; the integral operator T , which maps φ into another function, is defined by [T φ](p0) = −. The following simple lemma, on the growth of deviations in the successive backward iterates of the finite-dimensional p equation, is necessary in the proof of the proposition.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
