Bivariate spline interpolation with optimal approximation order
type: Item
instance of: scholarly article
title: Bivariate spline interpolation with optimal approximation order
zbMATH Open document ID: 1004.41005
author: O. Davydov
author: Günther Nürnberger
author: Frank Zeilfelder
published in: Constructive Approximation
publication date: 2003-02-11T00:00:00Z
full work available at URL: https://strathprints.strath.ac.uk/36669/
review text: Let \(S^r_q(\Delta)\) denote the space of bivariate polynomial splines of smoothness \(r\) and degree \(q\), with respect to a triangulation \(\Delta\) of a polygonal domain \(\Omega\subset\mathbb{R}^2\). The paper deals with some approximation properties of the superspline subspace \[ S^{r,\rho}_q(\Delta)= \{s\in S^r_q(\Delta)\mid s\in C^\rho(v),\;\forall\text{ vertices }v\text{ of }\Delta\}, \] where \(\rho= r+ [(r+ 1)/2]\) and \(q\geq 3r+2\). The main result of the paper establishes: A) the existence of a unique \(s_f\in S^{r,\rho}_q(\Delta)\) satisfying Hermite-type interpolation conditions for a given \(f\in C^{2r}(\Omega)\), B) the following estimate for \(f\in C^m(\Omega)\) \((m= 2r,\dots, q+1)\), and \(T\in\Delta\) \[ \|D^\alpha_x D^\beta_y(f- s_f)\|_{L_\infty(T)}\leq Kh^{m- \alpha-\beta}_T \max_{0\leq \mu\leq m}\|D^\mu_x D^{m- \mu}_y f\|_{C(T)} \] for all \(\alpha,\beta\geq 0\), \(\alpha+ \beta\leq m\), where \(h_T\) is the diameter of \(T\), and \(K= K(r,q,\theta_\Delta)\) (\(\theta_\Delta\) is the smallest angle in \(\Delta\)). The interpolation scheme is given in terms of a nodal basis \(\{s_i\}\) of \(S^{r,\rho}_q(\Delta)\), which is studied in the last section of the article. The authors point out that one of the several advantages of their approach, is that the estimation of the approximation error is completely local. It means that the estimate involves only the local Sobolev norm of \(f\) on the triangle \(T\) on which the error is measured.
reviewed by: J. Illán
Mathematics Subject Classification ID: 41A25
Mathematics Subject Classification ID: 41A63
Mathematics Subject Classification ID: 41A05
Mathematics Subject Classification ID: 41A15
Mathematics Subject Classification ID: 65D05
Mathematics Subject Classification ID: 65D07
zbMATH DE Number: 1618395
zbMATH Keywords: bivariate spline
zbMATH Keywords: interpolation
zbMATH Keywords: approximation order
zbMATH Keywords: minimally supported basis
zbMATH Keywords: locally linearly independent basis
zbMATH Keywords: Hermite-type interpolation
MaRDI profile type: Publication
OpenAlex ID: W2012790650
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DOI: 10.1007/S003650010034
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description: scientific article; zbMATH DE number 1618395