Approximation of the global attractor for the incompressible Navier-Stokes equations

This paper considers the asymptotic behaviour of a practical numerical approximation of the Navier-Stokes equations in Ω, a bounded subdomain of ℝ2. The scheme consists of a conforming finite element spatial discretization, combined with an order-preserving linearly implicit implementation of the second-order BDF method. It is shown that the method possesses a compact global attractor, which is upper semicontinuous with respect to the attractor of the underlying system in H1 (Ω). The proofs employ the techniques of G-stability, discrete Sobolev estimates for the Stokes operator similar to those of Heywood and Rannacher, semigroups of linear operators and attractor convergence theory in the context of multistep methods.