In this work, we introduced a class of nonlocal models to accurately approximate the Poisson model on manifolds that are embedded in high dimensional Euclid spaces with Dirichlet boundary. In comparison to the existing nonlocal Poisson models, instead of utilizing volumetric boundary constraint to reduce the truncation error to its local counterpart, we rely on the Poisson equation itself along the boundary to explicitly express the second order normal derivative by some geometry-based terms, so that to create a new model with $\mathcal{O}(\delta)$ truncation error along the $2\delta-$boundary layer and $\mathcal{O}(\delta^2)$ at interior, with $\delta$ be the nonlocal interaction horizon. Our concentration is on the construction and the truncation error analysis of such nonlocal model. The control on the truncation error is currently optimal among all nonlocal models, and is sufficient to attain second order localization rate that will be derived in our subsequent work.

翻译：本文提出了一类非局部模型，用于精确逼近嵌入高维欧几里得空间且具有Dirichlet边界的流形上的泊松模型。与现有非局部泊松模型相比，我们不再采用体积边界约束来将截断误差降至局部对应水平，而是利用边界上的泊松方程本身，通过一些几何项显式表达二阶法向导数，从而构建一个新模型，其在$2\delta-$边界层内具有$\mathcal{O}(\delta)$截断误差，在内部区域具有$\mathcal{O}(\delta^2)$截断误差，其中$\delta$为非局部相互作用范围。研究重点集中在此类非局部模型的构建与截断误差分析上。该截断误差控制水平在所有非局部模型中目前是最优的，足以达到二阶局部化速率，这一结果将在后续工作中推导得出。