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

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.