The divergence constraint of the incompressible Navier-Stokes equations is revisited in the mixed finite element framework. While many stable and convergent mixed elements have been developed throughout the past four decades, most classical methods relax the divergence constraint and only enforce the condition discretely. As a result, these methods introduce a pressure-dependent consistency error which can potentially pollute the computed velocity. These methods are not robust in the sense that a contribution from the right-hand side, which influences only the pressure in the continuous equations, impacts both velocity and pressure in the discrete equations. This talk reviews the theory and practical implications of relaxing the divergence constraint. Several approaches for improving the discrete mass balance or even for computing divergence-free solutions will be discussed: grad-div stabilization, higher order mixed methods derived on the basis of an exact de Rham complex, and mixed methods with an appropriate reconstruction of  the test functions. Numerical examples illustrate both the potential effects of using non-robust discretizations and the improvements obtained by utilizing pressure-robust  discretizations.
This topic is joint work with Alexander Linke (WIAS Berlin), Christian Merdon (WIAS Berlin), Michael Neilan (Pittsburgh), and Leo G. Rebholz (Clemson).

Professor Volker John obtained his PhD at the University of Magdeburg in 1997 and he defended his habilitation thesis in 2002. In 2005 he became full professor at the University of the Saarland in Saarbruecken. In 2009 he became the head of one of the research groups at the Weierstrass Institute for Applied Analysis and Stochastics in Berlin. This position is combined with a full professorship at the Free University of Berlin. Professor John's current research interests are stabilized discretizations for convection-dominated problems, finite element methods for incompressible flow problems, and application that require the numerical solution of the above mentioned problems.