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Determining nodes, finite differences schemes and inertial manifolds. (English) Zbl 0714.34078

Summary: The aim of this paper is to present a connection between the concepts of determining nodes and inertial manifolds with that of finite difference and finite volumes approximation to dissipative partial differential equations. In order to illustrate this connection we consider the one dimensional Kuramoto-Sivashinsky equation as an instructive paradigm. We remark that the results presented here apply to many other equations such as the one dimensional complex Ginzburg-Landau equation, the Chafee- Infante equation, etc.

MSC:

34C40 Ordinary differential equations and systems on manifolds
65M99 Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems
65N99 Numerical methods for partial differential equations, boundary value problems
35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
34K99 Functional-differential equations (including equations with delayed, advanced or state-dependent argument)
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
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