Local Lagrange Interpolation by Bivariate C 1 Cubic Splines

Lagrange interpolation schemes are constructed based on C 1 cubic splines on certain triangulations obtained from checkerboard quad-rangulations. x1. Introduction Given a triangulation 4 of a simply connected polygonal domain , the space of C 1 cubic splines is deened by S 1 3 (4) := fs 2 C 1 (() : sj T 2 P 3 , all T 2 4g; where P 3 is the space of cubic bivariate polynomials. In this paper we are interested in constructing spline interpolation methods that are based on a given set of Lagrange data and which deliver full approximation power. It is well known that to work with S 1 3 (4) successfully, we have to restrict our attention to special classes of triangulations. Indeed, for general triangulations, at this point it is not known whether interpolation at all of the vertices of 4 is even possible, and the dimension of S 1 3 (4) is also unknown. Moreover, it is known 3] that the space is defective in the sense that it does not give full approximation power on some triangulations (including the very regular type-I triangulations). This implies that in general it does not have a stable local basis. There are several classes of triangulations where the situation is sim-pliied. First, one can work on the reened triangulation 4 CT which is obtained from 4 by splitting each triangle into three subtriangles. The classical Clough-Tocher C 1 cubic element can then be constructed locally from values and gradients at each of the vertices of 4. If certain cross-derivative information is also available, the method gives full approximation power, see Mathematical Methods in CAGD: Oslo 2000 0 Tom Lyche and Larry L. Schumaker (eds.), pp. 0|1. ISBN 0-8265-xxxx-x. All rights of reproduction in any form reserved.