Mixed and hybrid finite element methods
Mixed and hybrid finite element methods
Numerical solution of the stationary Navier-Stokes equations by finite element methods
Proceedings of the International Symposium on Computing Methods in Applied Sciences and Engineering, Part 1
A Multigrid Method for the Pseudostress Formulation of Stokes Problems
SIAM Journal on Scientific Computing
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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.