We present a parametric finite element approximation of two-phase flow. This
free boundary problem is given by the Stokes equations in the two phases, which
are coupled via jump conditions across the interface. Using a novel variational
formulation for the interface evolution gives rise to a natural discretization of the
mean curvature of the interface. In addition, the mesh quality of the parametric
approximation of the interface does not deteriorate, in general, over time; and an
equidistribution property can be shown for a semidiscrete continuous-in-time variant
of our scheme in two space dimensions. Moreover, on using a simple XFEM pressure
space enrichment, we obtain exact volume conservation for the two phase regions.
Furthermore, our fully discrete finite element approximation can be shown to be
unconditionally stable. We demonstrate the applicability of our method with some
numerical results which, in particular, demonstrate that spurious velocities can be
avoided in the classical test cases.