Momentum and mass conservative mixed FEM for the Stokes problem

In this work, we analyze 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, while 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 numerical schemes that preserve mass and momentum conservation in the two-dimensional case, without increasing the computational cost, and mass conservation in the three-dimensional case. Finally, numerical experiments are presented to confirm the theoretical results.