Quadratic divergence-free finite elements on Powell–Sabin tetrahedral grids

Given a tetrahedral grid in 3D, a Powell–Sabin grid can be constructed by refining each original tetrahedron into 12 subtetrahedra. A new divergence-free finite element on 3D Powell–Sabin grids is constructed for Stokes equations, where the velocity is approximated by continuous piecewise quadratic polynomials while the pressure is approximated by discontinuous piecewise linear polynomials on the same grid. To be precise, the finite element space for the pressure is exactly the divergence of the corresponding space for the velocity. Therefore, the resulting finite element solution for the velocity is pointwise divergence-free, including the inter-element boundary. By establishing the inf-sup condition, the finite element is stable and of the optimal order. Numerical tests are provided.