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 continuousin-
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.