A finite element formulation for the Stokes problem allowing equal velocity-pressure interpolation

In this paper we study a variational formulation of the Stokes problem that accommodates the use of equal velocity-pressure finite element interpolations. The motivation of this method relies on the analysis of a class of fractional-step methods for the Navier-Stokes equations for which it is known that equal interpolations yield good numerical results. The reason for this turns out to be the difference between two discrete Laplacian operators computed in a different manner. The formulation of the Stokes problem considered here aims to reproduce this effect. From the analysis of the finite element approximation of the problem we obtain stability and optimal error estimates using velocity-pressure interpolations satisfying a compatibility condition much weaker than the inf-sup condition of the standard formulation. In particular, this condition is fulfilled by the most common equal order interpolations.