Stabilized finite element schemes for incompressible flow using Scott–Vogelius elements

We propose a stabilized mixed finite element method based on the Scott–Vogelius element for the Oseen equation. Here, only convection has to be stabilized since by construction both the discrete pressure and the divergence of the discrete velocities are controlled in the norm $L^2$. As stabilization we propose either the local projection stabilization or the interior penalty stabilization based on the penalization of the gradient jumps over element edges. We prove a discrete inf–sup condition leading to optimal a priori error estimates. Moreover, convergence of the velocities is completely independent of the pressure regularity, and in the purely incompressible case the discrete velocities are pointwise divergence free. The theoretical considerations are illustrated by some numerical examples