A Robust Numerical Method for the Stokes Equations Based on Divergence-free H (div) Finite Element Methods

A computational procedure based on a divergence-free H(div) method is presented for the Stokes and Navier-Stokes equations in this article. This method is designed to find velocity approximation in an exact divergence-free subspace of the corresponding H(div) finite element space. That is, the continuity equation is strongly enforced a priori and the pressure is eliminated from the calculation. A strength of this approach is that the saddle point problem for the Stokes equations is reduced to a symmetric positive definite problem in a subspace for which basis functions are readily available. The resulting discrete system can then be solved by using existing sophisticated solvers. The aim of this article is to demonstrate the efficiency and robustness of H(div) finite element methods for the Stokes and Navier-Stokes equations. The results not only confirm the existing theoretical predictions, but also reveal additional advantages of the method in dealing with discontinuous boundary conditions.