Two tetrahedral C1C1 cubic macro elements

We propose two tetrahedral C1C1 cubic macro elements that are constructed locally on one tetrahedron without any knowledge of the geometry of neighboring tetrahedra. Among such geometrically unconstrained local polynomial tetrahedral C1C1 schemes requiring only first order derivative data, our macro elements have the smallest number of coefficients. The resulting macro element spaces are stable and provide full approximation power. We give explicit formulae that can be used to implement our schemes.