MIXED FINITE ELEMENT METHODS FOR INCOMPRESSIBLE FLOW: STATIONARY NAVIER-STOKES EQUATIONS

In [Z. Cai, C. Tong, P. S. Vassilevski, and C. Wang, Numer. Methods Partial Differential Equations, to appear], the authors developed and analyzed a mixed finite element method for the stationary Stokes equations based on the pseudostress-velocity formulation. The pseudostress and the velocity are approximated by a stable pair of finite elements: Raviart-Thomas elements of index k >= 0 and discontinuous piecewise polynomials of degree k >= 0, respectively. This paper extends the method to the stationary, incompressible Navier-Stokes equations. Under appropriate assumptions, we show that the pseudostress-velocity formulation of the Navier-Stokes equation and its discrete counterpart have branches of nonsingular solutions, and error estimates of the mixed finite element approximations are established as well.