On the $P_1$ Powell-Sabin Divergence-Free Finite Element for the Stokes Equations

Source abstract

The stability of the $P_1$-$P_0$ mixed-element is established on general Powell-Sabin triangular grids. The piecewise linear finite element solution approximating the velocity is divergence-free pointwise for the Stokes equations. The finite element solution approximating the pressure in the Stokes equations can be obtained as a byproduct if an iterative method is adopted for solving the discrete linear system of equations. Numerical tests are presented confirming the theory on the stability and the optimal order of convergence for the $P_1$ Powell-Sabin divergence-free finite element method.