Smooth macro-elements based on Clough-Tocher triangle splits (Q1348927)

A bivariate macro-element defined on a triangle consists of a finite dimensional linear-space \(\mathcal S\) defined on the triangle and a set of linear functionals forming a basis for the dual of \(\mathcal S\). A macro-element defines a local interpolation scheme. The first \({\mathcal C}^r\) macro-elements were constructed using polynomials of degree \(4r+1\). To get macro-elements using lower degree polynomials, it is necesssary to split the triangle. In the present paper, macro-elements of smoothness \({\mathcal C}^r\) on Clough-Tocher triangle splits are constructed for all \(r\geq 0\). These new elements are improvements on elements constructed in a paper by \textit{M. J. Lai} and \textit{L. L. Schumaker} [ibid. 88, No.~1, 105-119 (2001; Zbl 0989.65013)]. The application of the new elements to Hermite interpolation of scattered data is described and numerical experiments are given.