Trivariate Local Lagrange Interpolation and Macro Elements of Arbitrary Smoothness

Michael A. Matt constructs two trivariate local Lagrange interpolation methods which yield optimal approximation order and C^r 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 second is the first trivariate Lagrange interpolation method using C² splines. It is based on arbitrary tetrahedral partitions using splines of degree nine. The author constructs trivariate macro-elements based on the Alfeld split, where each tetrahedron is divided into four subtetrahedra, and the Worsey-Farin split, where each tetrahedron is divided into twelve subtetrahedra, of a tetrahedral partition. In order to obtain the macro-elements based on the Worsey-Farin split minimal determining sets for C^r macro-elements are constructed over the Clough-Tocher split of a triangle, which are more variable than those in the literature.