Bivariate spline interpolation with optimal approximation order

Tools

Davydov, Oleg and Nurnberger, G. and Zeilfelder, F. (2001) Bivariate spline interpolation with optimal approximation order. Constructive Approximation, 17 (2). pp. 181-208. (https://doi.org/10.1007/s003650010034)

[thumbnail of 3r2a.pdf]
Preview
 PDF. Filename: 3r2a.pdf
Preprint
Download (336kB)| Preview

Abstract

Let be a triangulation of some polygonal domain f c R2 and let S9 (A) denote the space of all bivariate polynomial splines of smoothness r and degree q with respect to A. We develop the first Hermite-type interpolation scheme for S9 (A), q >_ 3r + 2, whose approximation error is bounded above by Kh4+i, where h is the maximal diameter of the triangles in A, and the constant K only depends on the smallest angle of the triangulation and is independent of near-degenerate edges and nearsingular vertices. Moreover, the fundamental functions of our scheme are minimally supported and form a locally linearly independent basis for a superspline subspace of Sr, (A). This shows that the optimal approximation order can be achieved by using minimally supported splines. Our method of proof is completely different from the quasi-interpolation techniques for the study of the approximation power of bivariate splines developed in [71 and [181.

CORE (COnnecting REpositories)