We propose a novel fitted finite element method for two-phase Stokes
flow problems that uses piecewise linear finite elements to approximate the
moving interface. The method can be shown to be unconditionally stable.
Moreover, spherical stationary solutions are captured exactly by the numerical
approximation. In addition, the meshes describing the discrete interface
in general do not deteriorate in time, which means that in numerical simulations
a smoothing or a remeshing of the interface mesh is not necessary.
We present several numerical experiments for our numerical method, which
demonstrate the accuracy and robustness of the proposed algorithm.