Generalized postprocessed approximations to the Navier–Stokes equations based on two grids

Abstract

In this paper we generalize previous postprocessed approximations to the Navier–Stokes equations. The idea of postprocessing in the context of mixed finite element approximations is the following. Once a standard Galerkin mixed finite element approximation is computed using a mesh (we will call coarse) of size H at a fixed time T, we postprocess the approximation using a finer mesh of size h<H, by solving a steady problem with data based on the Galerkin approximation. In the literature, one can find different ways to postprocess the Galerkin approximation depending on the problem that is solved over the fine mesh at the final time T. Basically, one can solve a Stokes problem, an Oseen type problem or a Newton type problem. In the present paper, we present a method based on several parameters that generalize the above postprocessing techniques. Depending on the values chosen for the different parameters one can recover one of the old (known) postprocessing procedures but also produce a new (different) method. We get error bounds for the generalized method valid for any of the values of the different parameters. In all the cases, the postprocessed method has a rate of convergence one unit bigger than the rate of convergence of the plain Galerkin method in terms of the coarse mesh H and optimal in terms of the fine mesh h. The computational added cost of postprocessing is however negligible. For the error analysis we do not assume nonlocal compatibility conditions for the true solutions of the Navier–Stokes equations. Some numerical experiments show the performance of the method for different values of the parameters.