Macro-elements of arbitrary smoothness over the Worsey-Farin split of a tetrahedron

Abstract

In this chapter we describe the Cr macro-elements based on the Worsey- Farin split of a tetrahedron (see Definition 2.7) by Matt [66]. In section 8.1, we review the minimal conditions for the degree of polynomials and the degree of supersmoothness for constructing Cr macro-elements over the Worsey-Farin split. In the following section 8.2, we investigate minimal determining sets for Cr macro-elements defined on the Worsey-Farin split of tetrahedra. In the next section, we illustrate these minimal determining sets with some examples to ease their understanding. In section 8.4, we examine nodal minimal determining sets for the Cr splines considered in this chapter. Finally, in section 8.5, a Hermite interpolant set for Cr splines based on the Worsey-Farin split is constructed. Moreover, it is shown that the interpolation yields optimal approximation order.