  We present a parametric finite element approximation of two-phase flow. This
free boundary problem is given by the Navier--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. The parametric
finite element approximation of the evolving interface is then coupled to a
standard finite element approximation of the two-phase Navier--Stokes equations
in the bulk. Here enriching the pressure approximation space with the help of
an XFEM function ensures good volume conservation properties for the two phase
regions. In addition, the mesh quality of the parametric approximation of the
interface in general does not deteriorate over time, and an equidistribution
property can be shown for a semidiscrete continuous-in-time variant of our
scheme in two space dimensions. Moreover, our finite element approximation can
be shown to be unconditionally stable. We demonstrate the applicability of our
method with some numerical results in two and three space dimensions.
