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.

翻译：广义重心坐标的梯度界在广义重心插值的 $H^1$ 范数逼近误差估计中起着至关重要的作用。类似地，$H^k$ 范数（$k>1$）的估计需要高阶导数的上界，而这在现有文献中尚不可得。本文针对简单凸 $d$ 维多面体（$d\ge 1$）上的 Wachspress 广义重心坐标，推导了此类高阶导数的上界。该结果可用于证明基于 Wachspress 坐标的多面体有限元方法在逼近（例如）四阶椭圆方程时具有最优收敛性。本文的另一贡献是比较了简单凸多面体的各种形状正则性条件，并利用凸几何知识阐明了它们之间的关系。