  We consider the discretization of a stationary Stokes interface problem in a
velocity-pressure formulation. The interface is described implicitly as the
zero level of a scalar function as it is common in level set based methods.
Hence, the interface is not aligned with the mesh. An unfitted finite element
discretization based on a Taylor-Hood velocity-pressure pair and an XFEM (or
CutFEM) modification is used for the approximation of the solution. This allows
for the accurate approximation of solutions which have strong or weak
discontinuities across interfaces which are not aligned with the mesh. To
arrive at a consistent, stable and accurate formulation we require several
additional techniques. First, a Nitsche-type formulation is used to implement
interface conditions in a weak sense. Secondly, we use the ghost penalty
stabilization to obtain an inf-sup stable variational formulation. Finally, for
the highly accurate approximation of the implicitly described geometry, we use
a combination of a piecewise linear interface reconstruction and a parametric
mapping of the underlying mesh. We introduce the method and discuss results of
numerical examples.
