Summary: This paper considers the asymptotic behaviour of a practical numerical approximation of Navier-Stokes equations in \(\Omega\), a bounded subdomain of \(\mathbb{R}^2\). 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 \(H^1(\Omega)\). The proofs employ the techniques of \(G\)-stability, discrete Sobolev estimates for Stokes operator, semigroups of linear operators, and attractor convergence theory in the context of multistep methods.