Approximation of the multi-dimensional Stokes system with embedded pressure discontinuities

Surface tension in multi-phase fluid flow engenders pressure discontinuities on phase interfaces. In this work we present a finite element method to solve viscous incompressible flows problems, especially designed to cope with such a situation. Taking as a model the Stokes system we study a finite element solution method based on a classical Galerkin-least-squares formulation with an added pressure jump term multiplied by the mesh step size. Both the velocity and the pressure are represented with continuous piecewise linear functions except for the latter field on the embedded interface. A suitable modification of the pressure space is employed in order to represent interface discontinuities. A priori error analyses point to optimal convergence rates for this approach.