  We consider the numerical approximation of a sharp-interface model for
two-phase flow, which is given by the incompressible Navier-Stokes equations in
the bulk domain together with the classical interface conditions on the
interface. We propose structure-preserving finite element methods for the
model, meaning in particular that volume preservation and energy decay are
satisfied on the discrete level. For the evolving fluid interface, we employ
parametric finite element approximations that introduce an implicit tangential
velocity to improve the quality of the interface mesh. For the two-phase
Navier-Stokes equations, we consider two different approaches: an unfitted and
a fitted finite element method, respectively. In the unfitted approach, the
constructed method is based on an Eulerian weak formulation, while in the
fitted approach a novel arbitrary Lagrangian-Eulerian (ALE) weak formulation is
introduced. Using suitable discretizations of these two formulations, we
introduce two finite element methods and prove their structure-preserving
properties. Numerical results are presented to show the accuracy and efficiency
of the introduced methods.
