In this talk we will present a postprocessing procedure for the Galerkin method which involves the use of an approximate inertial manifold to model the high wavenumbers component of the solution in terms of the low wavenumbers. This {\it postprocessing Galerkin method}, which is much cheaper to implement computationally than the Nonlinear Galerkin (NLG) Method, possess the same rate of convergence (accuracy) as the simplest version of the NLG, which is more accurate than the standard Galerkin method. Our results valid in the context of spectral and finite element Galerkin methods and for many nonlinear parabolic equations including the Navier-Stokes equations. We will also present some computational study to support our analytical results.

This talk is based on joint works with Bosco Garcia-Archilla, Len Margolin, Julia Novo and Shannon Wynne.