Two-Grid Mixed Finite-Element Approximations to the Navier-Stokes Equations Based on a Newton-Type Step

A two-grid scheme to approximate the evolutionary Navier-Stokes equations is introduced and analyzed. A standard mixed finite element approximation is first obtained over a coarse mesh of size H at any positive time . Then, the approximation is postprocessed by means of solving a steady problem based on one step of a Newton iteration over a finer mesh of size . The method increases the rate of convergence of the standard Galerkin method in one unit in terms of H and equals the rate of convergence of the standard Galerkin method over the fine mesh h. However, the computational cost is essentially the cost of approaching the Navier-Stokes equations with the plain Galerkin method over the coarse mesh of size H since the cost of solving one single steady problem is negligible compared with the cost of computing the Galerkin approximation over the full time interval (0, T]. For the analysis we take into account the loss of regularity at initial time of the solution of the Navier-Stokes equations in the absence of nonlocal compatibility conditions. Some numerical experiments are shown.