The Postprocessing Galerkin and Nonlinear Galerkin Methods: A Truncation Analysis Point of View

We revisit the postprocessing algorithm and give a justification from a classical truncation analysis point of view. We assume a perturbation expansion for the high frequency mode component of solutions to the underlying equation. Keeping terms to certain orders, we then generate approximate systems which correspond to numerical schemes. We show that the first two leading order methods are in fact the postprocessed Galerkin and postprocessed nonlinear Galerkin methods, respectively. Hence postprocessed Galerkin is a natural leading order method, more natural than the standard Galerkin method, for approximating solutions of parabolic dissipative PDEs. The analysis is presented in the framework of the two-dimensional Navier-Stokes equation (NSE); however, similar analysis may be done for any parabolic, dissipative nonlinear PDE. The truncation analysis is based on asymptotic estimates (in time) for the low and high mode components. We also introduce and investigate an alternative postprocessing scheme, which we call the dynamic postprocessing method, for the case in which the asymptotic estimates (in time) do not hold (i.e., in the situation of long transients, nonsmooth initial data, or highly oscillatory time-dependent solutions).