Construction of a $C^2$ class finite element based on the Clough-Tocher subdivision

In this paper, we construct a $C^2$ finite element based on the Clough-Tocher subdivision. We use derivatives order up to two at the vertices and cross boundary derivatives order up to two along the exterior edges of the triangle. The centroid of the triangle is just evaluated. The interpolant used is globally $C^2,$ has local support, is piecewise polynomial of degree less or equal to 5.