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

TL;DR

This paper derives upper bounds for high-order derivatives of Wachspress coordinates on convex polytopes, enabling better error estimates in finite element methods and clarifying shape-regularity conditions.

Contribution

It provides the first upper bounds for high-order derivatives of Wachspress coordinates on convex polytopes, facilitating optimal convergence analysis in finite element approximations.

Findings

Derived upper bounds for high-order derivatives of Wachspress coordinates.

Compared shape-regularity conditions and clarified their relations.

Enabled proof of optimal convergence for polytopal finite element methods.

Abstract

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.