Attractors and Inertial Manifolds for Finite Diierence Approximations of the Complex Ginzburg{landau Equation

A semi{discrete spatial nite diierence approximation and two fully discrete nite diierence approximations to the complex Ginzburg{Landau equation are considered in this paper. The existence of an inertial manifold is proved inside a discrete H 1 absorbing ball for the semi{discrete approximation by showing certain spectral properties hold for the linear operator and certain Lipschitz properties hold for the non{linear term. Convergence of the true and discrete inertial mani-folds is then shown by considering the contraction mappings used to construct the inertial manifolds. A fully implicit scheme is shown to form a continuous semi{group in a discrete L 2 space and absorbing balls of radii independent of the spatial and temporal mesh sizes in discrete L 2 and H 1 spaces are constructed; existence of a global attractor follows. Furthermore the existence of an inertial manifold is proved by showing certain spectral and Lipschitz properties hold. For a semi{ implicit scheme existence is proved of absorbing balls in the discrete space L 2 of radii independent of the spatial and temporal mesh; existence of a global attractor follows.