In this paper, we propose a class of nonlocal models to approximate the Poisson model on manifolds with homogeneous Neumann boundary condition, where the manifolds are assumed to be embedded in high dimensional Euclid spaces. In comparison to the existing nonlocal approximation of Poisson models with Neumann boundary, we optimize the truncation error of model by adding an augmented term along the $2\delta$ layer of boundary, with $2\delta$ be the nonlocal interaction horizon. Such term is formulated by the integration of the second order normal derivative of solution through the boundary, while the second order normal derivative is expressed as the difference between the interior Laplacian and the boundary Laplacian. The concentration of our paper is on the construction of nonlocal model, the well-posedness of model, and its second-order convergence rate to its local counterpart. The localization rate of our nonlocal model is currently optimal among all related works even for the case of high dimensional Euclid spaces.

翻译：本文针对嵌入高维欧几里得空间的流形，提出了一类用于逼近具有齐次Neumann边界条件的泊松模型的非局部模型。与现有具有Neumann边界的泊松模型非局部逼近相比，我们通过沿$2\delta$边界层（其中$2\delta$为非局部相互作用视界）添加增广项来优化模型的截断误差。该增广项由解的二阶法向导数沿边界的积分构成，而二阶法向导数则表示为内部拉普拉斯算子与边界拉普拉斯算子之差。本文重点聚焦于非局部模型的构建、模型的适定性以及其相对于局部模型达到二阶收敛速率。即使在处理高维欧几里得空间情形时，本文提出的非局部模型的局部化速率在当前相关研究中仍属最优。