Upper bounds of higher-order derivatives for Wachspress coordinates on polytopes

Abstract

The gradient bounds of generalized barycentric coordinates (GBCs) play an essential role in the $H^{1}$ norm error estimate of generalized barycentric interpolations (Gillette, Rand & Bajaj (2012) Error estimates for generalized barycentric interpolation. Adv. Comput. Math., 37, 417–439.). Similarly, an $H^{k}$ norm error estimate, $k>1$, requires upper bounds of higher-order derivatives. Due to the nonpolynomial nature of GBCs, existing techniques for proving the gradient bounds do not easily extend to higher-order cases. In this paper, we propose a new method for deriving upper bounds of higher-order derivatives for the Wachspress GBCs 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 fourth- or higher-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.