Local approximation from spline spaces on box meshes
type: Item
instance of: scholarly article
title: Local approximation from spline spaces on box meshes
zbMATH Open document ID: 1469.41001
author: Andrea Bressan
author: Tom Lyche
published in: Foundations of Computational Mathematics
publication date: 2021-07-14T00:00:00Z
full work available at URL: URN:NBN:no-86233
review text: Piecewise polynomial approximations -- especially if the choice of the very useful quasi-interpolation method with splines or otherwise (e.g., kernel based methods) is made -- are well-studied ways to approximate functions in many variables. In this paper, the famous univariate piecewise polynomials (splines) are generalised to the multivariate situation by using tensor-product splines (other extensions are available). Error estimates (i.e., more precisely, approximation orders) are in the particular focus of the authors in this article when quasi-interpolation is used and in particular when so-called boxed meshes are employed. Remarkably, the authors are able to provide both \textit{local} and \textit{global} error estimates and those in the very general context of Sobolev norms. In particular, error bounds of approximations and their derivatives are derived. In some special cases, those error bounds are exponentially decreasing with the degree of the piecewise polynomials increasing, otherwise they are asymptotic with respect to point distances becoming smaller as is normal in the classical way. The results are quite general as error estimates are provided in the \(L_p/L_q\) fashion, i.e., approximation errors and their partial derivatives in \(L_p\) are estimated in \(L_q\) norms. Some of the estimates are isotropic (independent of direction), some depend on directions (anisotropic).
Mathematics Subject Classification ID: 41A15
Mathematics Subject Classification ID: 41A25
Mathematics Subject Classification ID: 41A63
zbMATH DE Number: 7371334
zbMATH Keywords: approximation
zbMATH Keywords: spline spaces
zbMATH Keywords: box meshes
zbMATH Keywords: quasi-interpolants
zbMATH Keywords: local error bounds
zbMATH Keywords: global error bounds
zbMATH Keywords: anisotropic error bounds
zbMATH Keywords: reduced seminorms
zbMATH Keywords: isogeometric analysis
reviewed by: Martin D. Buhmann
MaRDI profile type: Publication
OpenAlex ID: W3041581947
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DOI: 10.1007/S10208-020-09467-8
description: scientific article; zbMATH DE number 7371334