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 function involving the second order normal derivative along the $2\delta$ layer of boundary, with $2\delta$ be the nonlocal interaction horizon. The 2nd 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：本文提出一类非局部模型来逼近具有齐次诺伊曼边界条件的流形上的泊松模型，其中假设流形嵌入在高维欧几里得空间中。与现有带诺伊曼边界的泊松模型非局部逼近方法相比，我们通过添加一个包含沿边界 $2\delta$ 层的二阶法向导数的增广函数来优化模型的截断误差，其中 $2\delta$ 为非局部相互作用视域。该二阶法向导数表示为内部拉普拉斯算子与边界拉普拉斯算子之差。本文重点在于非局部模型的构建、模型的适定性及其对相应局部模型的二阶收敛速率。即使在欧几里得高维空间情形下，我们非局部模型的局部化速率目前在所有相关工作中也是最优的。