Macro-elements of arbitrary smoothness over the Alfeld split of a tetrahedron

Abstract

In this chapter the trivariate C^r macro-elements of Lai and Matt [54] based on the Alfeld split of a tetrahedron (see Definition 2.6) are considered. In section 7.1, we investigate the minimal conditions for the polynomial degree and the degree supersmoothness for splines based on the Alfeld split. In the next section, we consider minimal determining sets for C^r macro-elements over the Alfeld split. In the following section 7.3, in order to ease the understanding of the constructed minimal determining sets, we give some examples for these. In section 7.4, we illustrate nodal minimal determining sets for the macro-elements. Finally, in section 7.5, we construct a Hermite interpolant based on the Alfeld split, which yields optimal approximation order.