Trivariate Local Lagrange Interpolation and Macro Elements of Arbitrary Smoothness, 2012

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.

Lagrange Interpolation on Type-4 Partitions.- Trivariate Lagrange Interpolation with C² Splines.-  C^r Macro-Element over the Clough-Tocher Split.- C^r Macro-Element over the Alfeld Split.- C^r Macro-Element over the Worsey Farin Split.

Dr. Michael A. Matt completed his doctoral thesis under the supervision of Prof. Dr. Günther Nürnberger at the Chair of Mathematics IV, University of Mannheim.

Two trivariate local Lagrange interpolation methods are constructed which yield optimal approximation order and Cr macro-elements based on the Alfeld and the Worsey-Farin split of a tetrahedral partition