Smooth Macro-Elements Based on Powell-Sabin Triangle Splits

Macro-elements of smoothness C r on Powell-Sabin triangle splits are constructed for all r 0. These new elements are improvements on elements constructed in 10] in that certain unneeded degrees of freedom have been removed. 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 degrees of freedom, are usually taken to be point evaluations of derivatives. 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 6.1. 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 six subtriangles as follows. Suppose the vertices of T are V := fv 1 ; v 2 ; v 3 g, that v T is a point inside of T, and that W := fw 1 ; w 2 ; w 3 g are points on the edges hv i ; v i+1 i.