An approximate inertial manifolds approach to postprocessing the Galerkin method for the Navier-Stokes equations

In a recent paper we have introduced a postprocessing procedure for the Gal erkin method for dissipative evolution partial differential equations with periodic boundary conditions. The postprocessing technique uses approximate inertial manifolds to approximate the high modes (the small scale componen ts) in the exact solutions in terms of the Galerkin approximations, which i n this case play the role of the lower modes (large scale components). This procedure can be seen as a defect-correction technique. But contrary to st andard procedures, the correction is computed only when the time evolution is completed. Here we extend these results to more realistic boundary condi tions. Specifically, we study in detail the two-dimensional Navier-Stokes e quations subject to homogeneous (nonslip) Dirichlet boundary conditions. We also discuss other equations, such as reaction-diffusion systems and the C ahn-Hilliard equations.