  In this work, we present a parametric finite element approximation of
two-phase Navier-Stokes flow with viscoelasticity. The free boundary problem is
given by the viscoelastic Navier-Stokes equations in the two fluid phases,
connected by jump conditions across the interface. The elasticity in the fluids
is characterised using the Oldroyd-B model with possible stress diffusion. The
model was originally introduced to approximate fluid-structure interaction
problems between an incompressible Newtonian fluid and a hyperelastic
neo-Hookean solid, which are possible limit cases of the model. We approximate
a variational formulation of the model with an unfitted finite element method
that uses piecewise linear parametric finite elements. The two-phase
Navier-Stokes-Oldroyd-B system in the bulk regions is discretised in a way that
guarantees unconditional solvability and stability for the coupled
bulk-interface system. Good volume conservation properties for the two phases
are observed in the case where the pressure approximation space is enriched
with the help of an XFEM function. We show the applicability of our method with
some numerical results.
