Mixed Finite Element Methods for Incompressible Flow: Stationary Navier–Stokes Equations

Abstract

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 $k\geq0$ and discontinuous piecewise polynomials of degree $k\geq0$, 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.