Interpolation by Cubic Splines on Triangulations

We describe an algorithm for constructing point sets which admit unique Lagrange and Hermite interpolation from the space S 1 3 (() of C 1 splines of degree 3 deened on a general class of triangulations. The triangulations consist of nested polygons whose vertices are connected by line segments. In particular, we have to determine the dimension of S 1 3 (() which is not known for arbitrary triangulations. Numerical examples are given. x1. Introduction In the literature, point sets which admit unique Lagrange and Hermite interpolation from spaces S r q (() of splines of degree q and smoothness r were constructed for crosscut partitions , in particular for 1-and 2-partitions. Results on the approximation order of these interpolation methods were also proved. (Because of space limitations, we refer to the references of our paper 5] in this volume.) Hermite interpolation schemes for S 1 q ((); q 5, where is an arbitrary triangulation, were given in 1, 3]. An inductive method for constructing Lagrange and Hermite interpolation points for S 1 q ((); q 5, where is an arbitrary triangulation, was developed in 2]. Here, in each step, one vertex is added to the subtriangulation considered before. For q = 4, this method works under certain assumptions on. The most complex case is q = 3, since even the dimension of S 1 3 (() is not known for arbitrary triangulations. In this paper, we develop Lagrange and Hermite interpolation methods for S 1 3 ((). The triangulations consist of nested polygons whose vertices are connected in a natural way. The interpolation points are constructed inductively by passing through the vertices of the nested polygons, where in contrast to 2], the choice of these vertices is unique. All rights of reproduction in any form reserved.