Towards a Unified Finite Element Method for the Stokes Equations

We present an algorithm for solving the Stokes system based on the Scott--Vogelius technique together with the iterated penalty method. We show that by projecting the Scott--Vogelius pressure onto a continuous function space we can recover an accurate, continuous pressure solution while retaining a divergence-free velocity. This algorithm could be of particular interest when modeling incompressible complex fluids where there is a strong dependence on the pressure. We also show that it is more efficient than the Taylor--Hood approach in some cases. Full details of our implementation in the automated finite element software FEniCS are described, and numerical results are given.