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. Moreover a suitable modification of the pressure space is employed in order to represent interface discontinuities. A priori error analyses point to optimal convergence rates for both approaches, which justify observations from previous numerical experiments carried out in [3].