Upper bound of high-order derivatives for Wachspress coordinates on polytopes

The gradient bounds of generalized barycentric coordinates play an essential role in the \(H^1\) norm approximation error estimate of generalized barycentric interpolations. Similarly, the \(H^k\) norm, \(k>1\), estimate needs upper bounds of high-order derivatives, which are not available in the...

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Bibliographic Details
Published inarXiv.org
Main Authors Tian, Pengjie, Wang, Yanqiu
Format Paper
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 06.11.2024
Subjects
Approximation
Elliptic functions
Polytopes
Upper bounds
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Summary:The gradient bounds of generalized barycentric coordinates play an essential role in the \(H^1\) norm approximation error estimate of generalized barycentric interpolations. Similarly, the \(H^k\) norm, \(k>1\), estimate needs upper bounds of high-order derivatives, which are not available in the literature. In this paper, we derive such upper bounds for the Wachspress generalized barycentric coordinates on simple convex \(d\)-dimensional polytopes, \(d\ge 1\). The result can be used to prove optimal convergence for Wachspress-based polytopal finite element approximation of, for example, fourth-order elliptic equations. Another contribution of this paper is to compare various shape-regularity conditions for simple convex polytopes, and to clarify their relations using knowledge from convex geometry.
ISSN:2331-8422