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

Macro-elements of arbitrary smoothness are constructed on Clough-Tocher tri angle splits. These elements can be used for solving boundary-value problem s or for interpolation of Hermite data, and are shown to be optimal with re spect to spline degree. We conjecture they are also optimal with respect to the number of degrees of freedom. The construction provides local bases fo r certain superspline spaces defined over Clough-Tocher refinements of arbi trary triangulations. These bases are shown to be stable as a function of t he smallest angle in the triangulation, which in turn implies that the asso ciated spline spaces have optimal order approximation power.