2016, Ahmed, Naveed, John, Volker, Matthies, Gunar, Novo, Julia

The local projection stabilization (LPS) method in space is considered to approximate the evolutionary Oseen equations. Optimal error bounds independent of the viscosity parameter are obtained in the continuous-in-time case for the approximations of both velocity and pressure. In addition, the fully discrete case in combination with higher order continuous Galerkin-Petrov (cGP) methods is studied. Error estimates of order k + 1 are proved, where k denotes the polynomial degree in time, assuming that the convective term is time-independent. Numerical results show that the predicted order is also achieved in the general case of time-dependent convective terms.

2015, Frutos, Javier de, García-Archilla, Bosco, John, Volker, Novo, Julia

The approximation of the time-dependent Oseen problem using inf-sup stable mixed finite elements in a Galerkin method with grad-div stabilization is studied. The main goal is to prove that adding a grad-div stabilization term to the Galerkin approximation has a stabilizing effect for small viscosity. Both the continuous-in-time and the fully discrete case (backward Euler method, the two-step BDF, and CrankNicolson schemes) are analyzed. In fact, error bounds are obtained that do not depend on the inverse of the viscosity in the case where the solution is sufficiently smooth. The bounds for the divergence of the velocity as well as for the pressure are optimal. The analysis is based on the use of a specific Stokes projection. Numerical studies support the analytical results

2016, Ahmed, Naveed, Matthies, Gunar

We present in this paper numerical studies of higher order variational time stepping schemes combined with finite element methods for simulations of the evolutionary Navier-Stokes equations. In particular, conforming inf-sup stable pairs of finite element spaces for approximating velocity and pressure are used as spatial discretization while continuous GalerkinPetrov methods (cGP) and discontinuous Galerkin (dG) methods are applied as higher order variational time discretizations. Numerical results for the well-known problem of incompressible flows around a circle will be presented.