Local projection stabilization for the Oseen problem

We consider conforming finite element approximations of the time-dependent Oseen problem with inf-sup stable approximation of velocity and pressure. The work serves as a preliminary study of the incompressible Navier-Stokes problem. In the case of high Reynolds numbers, the local projection stabilization method is considered. In particular, the idea of streamline upwinding is combined with stabilization of the divergence-free constraint. For the arising semidiscrete problem, a stability and convergence analysis is given. Our approach improves some results of a recent paper by Matthies & Tobiska (2014, IMA J. Numer. Anal., 35, 239-269). Finally, we apply the approach to the time-dependent incompressible Navier-Stokes problem, test the accuracy of the method and conduct numerical experiments with simple boundary layers and separation.