Postprocessing the Galerkin method: A novel approach to approximate inertial manifolds

A postprocess of the standard Galerkin method for the discretization of dissipative equations is presented. The postprocessed Galerkin method uses the same approximate inertial manifold Phi(app) to approximate the high wave number modes of the solution as in the nonlinear Galerkin (NLG) method. However, in this postprocessed Galerkin method the value of Phi(app) is calculated only once, after the time integration of the standard Galerkin method is completed, contrary to the NLG in which Phi(app) evolves with time and affects the time evolution of the lower wave number modes. The postprocessed Galerkin method, which is much cheaper to implement computationally than the NLG, is shown, in the case of Fourier modes, to possess the same rate of convergence (accuracy) as the simplest version of the NLG, which is based on either the Foias-Manley-Temam approximate inertial manifold or the Euler-Galerkin approximate inertial manifold. This is proved for some problems in one and two spatial dimensions, including the Navier-Stokes equations under periodic boundary conditions. The advantages of postprocessing that we present here apply not only to the standard Galerkin method, but also to the computationally more efficient pseudospectral method.