Macro-elements and stable local bases for splines on Clough-Tocher triangulations

Macro-elements of arbitrary smoothness are constructed on Clough-Tocher triangle splits. These elements can be used for solving boundary-value problems or for interpolation of Hermite data, and are shown to be optimal with respect to spline degree. We believe they are also optimal with respect to the number of degrees of freedom. The construction provides local bases for certain superspline spaces deened over Clough-Tocher reenements of arbitrary triangulations. These bases are shown to be stable as a function of the smallest angle in the triangulation, which in turn implies that the associated spline spaces have optimal order approximation power. x1. Introduction Let 4 be triangulation of a polygonal domain in IR 2. In this paper we are interested in polynomial spline spaces of the form S r d (4) := fs 2 C r (() : sj T 2 P d for all T 2 4; g where d > r > 0 are given integers and P d is the space of bivariate polynomials of degree d. A basis fB i g n i=1 for a spline space S is called a stable local basis provided that there exist constants`; K 1 ; K 2 depending only on the smallest angle in 4 such that 1) for each 1 i n, there is a vertex v i of 4 for which supp(B i) star`(v i), 2) for all choices of the coeecient vector c = (c 1