Smooth macro-elements based on Clough-Tocher triangle splits

Macro-elements of smoothness C r on Clough-Tocher triangle splits are constructed for all r 0. These new elements are improvements on elements constructed in 11] in that (disproving a conjecture made there) certain unneeded degrees of freedom have been removed. Numerical experiments on Hermite interpolation with the new elements are included. A bivariate macro-element deened on a triangle T consists of a nite dimensional linear space S deened on T and a set of linear functionals forming a basis for the dual of S. It is common to choose the space S to be a space of polynomials or a space of piecewise polynomials deened on some subtriangulation of T. The members of , called the degrees of freedom, are usually taken to be point evaluations of derivatives, although here we will also work with sets of linear functionals which pick oo certain spline coeecients. A macro-element deenes a local interpolation scheme. In particular, if f is a suuciently smooth function, then we can deene the corresponding interpolant as the unique function s 2 S such that s = f for all 2. We say that a macro-element has smoothness C r provided that if the element is used to construct an interpolating function locally on each triangle of a triangulation 4, then the resulting piecewise function is C r continuous globally. The rst C r macro-elements were constructed using polynomials of degree 4r+1, see Remark 9.1. To get macro-elements using lower degree polynomials, it is necessary to split the triangle. Here we focus on the case where T is split into three subtriangles by connecting its vertices to some point v T in the interior. We call the resulting triangulation T CT the Clough-Tocher (CT) split of T. The standard choice for the interior vertex is the barycenter of T.