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

This is post-peer-review, pre-copyedit version of an article published in Journal of Scientific Computing. The final of authenticated version is available online at: https://doi.org/10.1007/s10915-017-0447-2A 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 T>0 . 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 h<H . 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 shownJ. Novo: Research supported by Spanish MINECO under grants MTM2013-42538-P (MINECO, ES) and MTM2016-78995-P (AEI/FEDER, UE

Finite elements for scalar convection-dominated equations and incompressible flow problems - A never ending story?

The contents of this paper is twofold. First, important recent results concerning finite element methods for convection-dominated problems and incompressible flow problems are described that illustrate the activities in these topics. Second, a number of, in our opinion, important problems in these fields are discussed