Optimal Triangulations for the Best C 1

In this paper, we are concerned with the problem of constructing triangulations which are optimal in the sense that the space of C 1 piecewise quartic polynomial functions on such triangulations admits the optimal ((fth) order of approximation. x1 Introduction Deenition 1.1. A collection = f i g N i=1 of triangles i in l R 2 is called a triangulation, if (i) the union := N S i=1 i is a connected set, and (ii) the intersection of any two adjacent triangles in is either a common vertex or a common edge. For a given triangulation with vertex set V , let V I denote the set of all interior vertices in , and V b := V n V I , the set of all boundary vertices in. We also denote the collection of all edges in by E and the collection of all interior edges in by E I. As usual, the notation S r k (() is used to denote the subspace of C r (() of all pp (:= piecewise polynomial) functions with total degree k and with grid lines given by the edges of. The choice of triangulations plays an important role in working with pp functions (also called splines). In 5], the problem of constructing triangu-lations which are optimal for some optimality criteria was studied. In this paper, we are concerned with optimal approximation order with respect to the given order r of smoothness and degree k of the polynomial pieces of the smooth pp functions. In other words, by optimal triangulation, we mean that the space of pp functions with degree k and smoothness order r on this triangulation achieves the highest order of approximation. Of