发表机构
University of Calabria（卡拉布里亚大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出适配环面几何的紧支撑多节点谢泼德算子，结合局部多项式插值与紧支撑权重实现环面上散乱数据插值，经数值实验验证其收敛性与适用性。
AI 中文摘要
我们引入一种紧支撑多节点谢泼德算子，用于嵌入在$\boldsymbol{\rm I\negthinspace R}^3$中的环面上散乱数据的插值。该方法将总次数为$d \boldsymbol{\rm \negthinspace \negthinspace N}$的局部多项式插值与紧支撑谢泼德型权重相结合，使得每个评估点处的近似值仅依赖于相邻的节点模板。环面被视为由四次多项式定义的代数曲面，我们使用格罗比纳基（Gröbner bases）构造该曲面上的约化多项式空间，以消除定义方程引起的冗余。这产生了适配环面几何的局部范德蒙德系统。我们讨论了环面的度量结构，证明了周期参数距离与从$\boldsymbol{\rm I\negthinspace R}^3$继承的欧氏距离之间的局部等价性，并利用该等价性为紧支撑构造提供依据。我们建立了关于最大支撑半径和局部勒贝格常数的一致误差估计。在均匀局部性和稳定性假设下，该方法关于填充距离的收敛阶为$d+1$。对解析测试函数的数值实验证实了多项式再生性，并呈现出与理论分析一致的误差衰减。该方法还在映射到环面的计算流体动力学数据上进行了测试，包括速度分量的插值、从插值分量重建速度大小，以及通过对插值分量的正交提升重建切向速度场。
英文摘要
We introduce a compactly supported multinode Shepard operator for the interpolation of scattered data on the torus embedded in $\mathbb{R}^3$. The method combines local polynomial interpolation of total degree $d\in \mathbb{N}$ with compactly supported Shepard-type weights, so that the approximation at each evaluation point depends only on neighbouring stencils of nodes. The torus is treated as an algebraic surface defined by a quartic polynomial, and Gröbner bases are used to construct reduced polynomial spaces on the surface by removing the redundancy induced by the defining equation. This yields local Vandermonde systems adapted to the toroidal geometry. We discuss the metric structure of the torus, show the local equivalence between the periodic parameter distance and the Euclidean distance inherited from $\mathbb{R}^3$, and use this equivalence to motivate the compact support construction. We establish a uniform error estimate in terms of the maximal support radius and the local Lebesgue constants. Under uniform locality and stability assumptions, the method converges with order $d+1$ with respect to the fill distance. Numerical experiments on analytical test functions confirm polynomial reproduction and exhibit an error decay consistent with the theoretical analysis. The approach is also tested on Computational Fluid Dynamics data mapped onto the torus, including the interpolation of the velocity components, the reconstruction of the velocity magnitude from the interpolated components, and the reconstruction of a tangent velocity field through an orthonormal lifting of the interpolated components.
Comments35 pages, 10 Figures
