  The paper addresses stability and finite element analysis of the stationary
two-phase Stokes problem with a piecewise constant viscosity coefficient
experiencing a jump across the interface between two fluid phases. We first
prove a priori estimates for the individual terms of the Cauchy stress tensor
with stability constants independent of the viscosity coefficient. Next, this
stability result is extended to the approximation of the two-phase Stokes
problem by a finite element method. In the method considered, the interface
between the phases does not respect the underlying triangulation, which put the
finite element method into the class of unfitted discretizations. The finite
element error estimates are proved with constants independent of viscosity.
Numerical experiments supporting the theoretical results are provided.
