Finite element methods for the Stokes system with interface pressure discontinuities

Surface tension in multi phase fluid flow engenders pressure discontinuities on phase interfaces. In this work we present two finite element methods to solve viscous incompressible flows problems, especially designed to cope with such a situation. Taking as a model the two-dimensional Stokes system, we consider solution methods based on piecewise linear approximations of both the velocity and pressure, with either velocity bubble or penalty enrichment, in order to obtain stable discrete problems. The novel ingredient is a suitable modification of the pressure space that is able to represent interface discontinuities. A priori error analyses are performed that point to optimal convergence rates for both approaches, thus justifying observations from previous numerical experiments reported in the literature. Some general results are also reported on the need for stabilization in the interface elements.