A General Approximation Scheme for Attractors of Abstract Parabolic Problems

ISSN： 0163-0563

Source： Numerical Functional Analysis and Optimization, Vol.27, Iss.7-8, 2006-12, pp. : 785-829

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Abstract

We consider semilinear problems of the form u ′ = Au + f ( u ), where A generates an exponentially decaying compact analytic C 0 -semigroup in a Banach space E , f : E α → E is differentiable globally Lipschitz and bounded ( E α = D ((− A ) α ) with the graph norm). Under a very general approximation scheme, we prove that attractors for such problems behave upper semicontinuously. If all equilibrium points are hyperbolic, then there is an odd number of them. If, in addition, all global solutions converge as t → ±∞, then the attractors behave lower semicontinuously. This general approximation scheme includes finite element method, projection and finite difference methods. The main assumption on the approximation is the compact convergence of resolvents, which may be applied to many other problems not related to discretization.