Local Mass-Corrections for Continuous Pressure Approximations of Incompressible Flow

In this work, we discuss a family of finite element discretizations for the incompressible Stokes problem using continuous pressure approximations on simplicial meshes. We show that after a simple and cheap correction, the mass-fluxes obtained by the considered schemes preserve local conservation on dual cells without reducing the convergence order. This allows the direct coupling to vertex-centered finite volume discretizations of transport equations. Further, we can postprocess the mass fluxes independently for each dual box to obtain an elementwise conservative velocity approximation of optimal order that can be used in cell-centered finite volume or discontinuous Galerkin schemes. Numerical examples for stable and stabilized methods are given to support our theoretical findings. Moreover, we demonstrate the coupling to vertexand cell-centered finite volume methods for advective transport.