Trivariate Local Lagrange Interpolation and Macro Elements of Arbitrary Smoothness

In this work, we construct two trivariate local Lagrange interpolation methods which yield optimal approximation order and Cr macro-elements based on the Alfeld and the Worsey-Farin split of a tetrahedral partition. The first interpolation method is based on cubic C1 splines over type-4 cube partitions, for which numerical tests are given. The other one is the first trivariate Lagrange interpolation method using C2 splines. It is based on arbitrary tetrahedral partitions using splines of degree nine. We construct trivariate macro-elements based on the Alfeld, where each tetrahedron is divided into four subtetrahedra, and the Worsey-Farin split, where each tetrahedron is divided into twelve subtetrahedra, of a tetra-hedral partition. In order to obtain the macro-elements based on the Worsey-Farin split we construct minimal determining sets for Cr macro-elements over the Clough-Tocher split of a triangle, which are more variable than those in the literature.