Recently, we constructed a class of nonlocal Poisson model on manifold under Dirichlet boundary with global $\mathcal{O}(\delta^2)$ truncation error to its local counterpart, where $\delta$ denotes the nonlocal horizon parameter. In this paper, the well-posedness of such manifold model is studied. We utilize Poincare inequality to control the lower order terms along the $2\delta$-boundary layer in the weak formulation of model. The second order localization rate of model is attained by combining the well-posedness argument and the truncation error analysis. Such rate is currently optimal among all nonlocal models. Besides, we implement the point integral method(PIM) to our nonlocal model through 2 specific numerical examples to illustrate the quadratic rate of convergence on the other side.

翻译：最近，我们在狄利克雷边界条件下构建了一类流形上的非局部泊松模型，该模型对其局部对应模型具有全局$\mathcal{O}(\delta^2)$截断误差，其中$\delta$表示非局部水平参数。本文研究了此类流形模型的适定性。我们利用庞加莱不等式来控制模型弱形式中沿$2\delta$边界层的低阶项。通过结合适定性论证与截断误差分析，得到了模型的二阶局部化率。该率是目前所有非局部模型中的最优结果。此外，我们通过两个具体数值算例，将点积分方法（point integral method, PIM）应用于非局部模型，从另一角度验证了二次收敛率。