A locally modified finite element method for a Stokes interface problem

In this work, we analyze a stationary Stokes interface problem. For discretization we apply locally modified second-order finite elements for the velocities combined with piecewise constant elements for pressure. The locally modified second-order finite element method is based on a fixed structured coarse mesh, which is then internally resolved and adjusted to the interface by means of a reference element mapping. This corresponds to a sub-triangulation of the coarse mesh into (possibly) anisotropic triangles. We show the stability of the $ \mathcal{P}_{2}\, -\, \mathcal{P}_{0} $ elements by using the macroelement technique, which requires local stability and a relatively weak global stability. In one particular case, we need to add a local stabilization term, or alternatively to move a critical vertex of the mesh by a small $ \epsilon $. Furthermore, we prove optimal error estimates in the energy norm and the $ L^2 $-norm of the velocity and show detailed numerical results.