Local Lagrange interpolation by bivariate C 1 cubic splines

Abstract

. Lagrange interpolation schemes are constructed based on C 1 cubic splines on certain triangulations obtained from checkerboard quadrangulations. x1. Introduction Given a triangulation 4 of a simply connected polygonal domain\\Omega\\Gamma the space of C 1 cubic splines is defined by S 1 3 (4) := fs 2 C 1 (\\Omega\\Gamma : 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 appr...