A momentum- and mass-conservative pseudostress-based mixed finite element method for the Stokes problem

In this paper we analyse a pseudostress-based mixed finite element method for the Stokes problem that ensures both mass and momentum conservation. Mass conservation is achieved by approximating the velocity with the lowest-order Raviart–Thomas elements, and momentum conservation is enforced through a discrete Helmholtz decomposition of the piecewise-constant vector space. We establish the well-posedness of the method and derive optimal convergence rates, including a superconvergence result for the velocity gradient and pressure approximations. In addition, to improve the efficiency of the method in terms of the total number of degrees of freedom, we propose equivalent pseudostress–stream-function finite element schemes in two and three dimensions that also preserve mass and momentum conservation. Finally, numerical experiments are presented to validate the theoretical results.