Simplex-splines on the Clough-Tocher element

We propose a simplex spline basis for a space of C1-cubics on the Clough-Tocher split on a triangle. The 12 elements of the basis give a nonnegative partition of unity. We derive two Marsden-like identities, three quasi-interpolants with optimal approximation order and prove L∞ stability of the basis. The conditions for C1-junction to neighboring triangles are simple and similar to the C1 conditions for the cubic Bernstein polynomials on a triangulation. The simplex spline basis can also be linked to the Hermite basis to solve the classical interpolation problem on the Clough-Tocher split.