标题： 在$\mathbb{R}^d$上受保护的Delaunay网格上梯度和矢量场的准最优插值 显示英文标题

标题： Quasi-optimal interpolation of gradients and vector-fields on protected Delaunay meshes in $\mathbb{R}^d$

摘要： 在多于二维的Delaunay网格上，关于函数或其梯度插值的数学结果非常少。 不幸的是，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，现有的结果主要是针对分段线性多项式的插值。 事实上，我们不知道有任何关于Delaunay网格上函数或其梯度的高阶、分段多项式插值的结果。 为了解决这个问题，我们证明了在受保护的Delaunay网格上可以成功实现准最优的、高阶的、分段多项式梯度插值。 此外，我们将分析推广到梯度插值之外，证明了足够光滑的向量场的准最优插值性质。 在整个论文中，我们使用“准最优”这个词，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，在受保护的Delaunay网格中，最小厚度可以在$\mathbb{R}^d$中被精确控制。 此外，目前关于最小厚度的最佳数学估计是在这种网格上获得的。 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一种具有更好最小厚度控制的替代Delaunay网格策略。 考虑到这一点，我们将受保护Delaunay网格上的插值称为准最优。 显示英文摘要

摘要： 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.