On C2 Quintic Spline Functions Over Triangulations of Powell-Sabin's Type

Given a triangulation 4 of a polygonal domain, we nd a reenement of 4 by choosing u t in a neighborhood of the center of the inscribed circle of each triangle t 2 4, connecting u t to the vertices of the triangle t, and connecting u t to u t 0 if t 0 2 4 shares an interior edge with t or to the midpoint v e of any boundary edge e of t. The resulting triangulation is a triangulation of Powell-Sabin's type. We investigate a C 2 quintic spline space ^ S 2 5 (() whose elements are C 3 only at u t 's. We give a dimension formula for this spline space, show how to construct a locally supported basis, display an interpolation scheme, and prove that this spline space has the full approximation order. x1. Introduction We are interested in constructing a C 2 smooth piecewise polynomial surface over a polygonal domain which interpolates or approximates a given set of scattered data over. That is, for given scattered data Schemes for constructing C 2 interpolatory surfaces have important applications in, e.g., the design of aircraft. There are several schemes already available in the literature. In general , there are two approaches based on the triangulation 4 of the set of scattered data: one is to construct an interpolating (or approximating) spline function of degree 8 or higher over 4 (see Chui and Lai'90] and Rescorla'86]); the other is to subdivide 4 and then construct a spline function of degree as low as 5 over the reenement of 4. Let S r d (4) := fs 2 C r (() : sj t 2 IP d ; 8t 2 4g be the spline function space of smoothness r and degree d and S r;; d (4) := fs 2 S r d (4) : s 2 C at each vertex of 4g: