Approximation of the global attractor for the incompressible Navier-Stokesequations

This paper considers the asymptotic behaviour of a practical numerical appr oximation of the Navier-Stokes equations in Omega, a bounded subdomain of R -2. The scheme consists of a conforming finite element spatial discretizati on, combined with an order-preserving linearly implicit implementation of t he second-order BDF method. it is shown that the method possesses a compact global attractor, which is upper semicontinuous with respect to the attrac tor of the underlying system in H-1 (Omega). The proofs employ the techniqu es of G-stability, discrete Sobolev estimates for the Stokes operator simil ar to those of Heywood and Rannacher, semigroups of linear operators and at tractor convergence theory in the context of multistep methods.