Mixed methods for stationary Navier-Stokes equations based on pseudostress-pressure-velocity formulation

Abstract

In this paper, we develop and analyze mixed finite element methods for the Stokes and Navier-Stokes equations. Our mixed method is based on the pseudostress-pressure-velocity formulation. The pseudostress is approximated by the Raviart-Thomas, Brezzi-Douglas-Marini, or Brezzi-Douglas-Fortin-Marini elements, the pressure and the velocity by piecewise discontinuous polynomials of appropriate degree. It is shown that these sets of finite elements are stable and yield optimal accuracy for the Stokes problem. For the pseudostress-pressure-velocity formulation of the stationary Navier-Stokes equations, the well-posedness and error estimation results are established. By eliminating the pseudostress variables in the resulting algebraic system, we obtain cell-centered finite volume schemes for the velocity and pressure variables that preserve local balance of momentum.