An AIM and one-step Newton method for the Navier–Stokes equations Original Research Article

In this paper, we investigate a two-level finite element approximation to the Navier–Stokes equations by means of a new approximate inertial manifold (AIM). Then we construct a new AIM-based numerical scheme and show that the convergence rate of the new approximation obtained by this AIM method is better than the double of the convergence rate of the standard Galerkin finite element solution. In addition, we also show that the new AIM scheme is equivalent to a one-step Newton iterative scheme for the Navier–Stokes equations in a suitable Hilbert space.