  This paper presents and analyzes an immersed finite element (IFE) method for
solving Stokes interface problems with a piecewise constant viscosity
coefficient that has a jump across the interface. In the method, the
triangulation does not need to fit the interface and the IFE spaces are
constructed from the traditional $CR$-$P_0$ element with modifications near the
interface according to the interface jump conditions. We prove that the IFE
basis functions are unisolvent on arbitrary interface elements and the IFE
spaces have the optimal approximation capabilities, although the proof is
challenging due to the coupling of the velocity and the pressure. The stability
and the optimal error estimates of the proposed IFE method are also derived
rigorously. The constants in the error estimates are shown to be independent of
the interface location relative to the triangulation. Numerical examples are
provided to verify the theoretical results.
