There are very few mathematical results governing the interpolation of functions or their gradients on Delaunay meshes in more than two dimensions. Unfortunately, the standard techniques for proving optimal interpolation properties are often limited to triangular meshes. Furthermore, the results which do exist, are tailored towards interpolation with piecewise linear polynomials. In fact, we are unaware of any results which govern the high-order, piecewise polynomial interpolation of functions or their gradients on Delaunay meshes. In order to address this issue, we prove that quasi-optimal, high-order, piecewise polynomial gradient interpolation can be successfully achieved on protected Delaunay meshes. In addition, we generalize our analysis beyond gradient interpolation, and prove quasi-optimal interpolation properties for sufficiently-smooth vector fields. Throughout the paper, we use the words 'quasi-optimal', because the quality of interpolation depends (in part) on the minimum thickness of simplicies in the mesh. Fortunately, the minimum thickness can be precisely controlled on protected Delaunay meshes in $\mathbb{R}^d$. Furthermore, the current best mathematical estimates for minimum thickness have been obtained on such meshes. In this sense, the proposed interpolation is optimal, although, we acknowledge that future work may reveal an alternative Delaunay meshing strategy with better control over the minimum thickness. With this caveat in mind, we refer to our interpolation on protected Delaunay meshes as quasi-optimal.

翻译：在超过二维的情况下，关于Delaunay网格上函数或其梯度插值的数学结果非常稀少。遗憾的是，证明最优插值性质的标准技术通常仅限于三角形网格。此外，现有的结果主要针对分段线性多项式的插值。事实上，我们尚未发现任何关于Delaunay网格上函数或其梯度的高阶分段多项式插值的结果。为解决这一问题，我们证明了在受保护的Delaunay网格上可以实现拟最优的高阶分段多项式梯度插值。此外，我们将分析推广至梯度插值之外，证明了足够光滑向量场的拟最优插值性质。在整篇论文中，我们使用“拟最优”一词，是因为插值的质量（部分）取决于网格中单纯形的最小厚度。幸运的是，在$\mathbb{R}^d$中的受保护Delaunay网格上，最小厚度可以被精确控制。而且，当前关于最小厚度的最佳数学估计正是在此类网格上获得的。从这个意义上说，所提出的插值是最优的，尽管我们承认未来的工作可能会揭示出一种能更好控制最小厚度的替代Delaunay网格生成策略。考虑到这一前提，我们将受保护Delaunay网格上的插值称为拟最优的。