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Drichlet边界下,与全球$\mathcal{O}(\ delta2\2) 建立了一组非本地的 Poisson 模型,用于Drichlet 边界下方的多元体, 与全球$\ mathcal{O}(\ delta2}2) 的脱轨错误对当地对应方, 其中$\delta$表示非本地地平线参数。 在本文中, 研究了这种多元模型的稳妥性。 我们用poincare 不平等来控制在2\ delta$- 边界层的低排序条件。 模型的第二顺序本地化率是通过将稳妥性参数和脱轨错误分析相结合来实现的。 目前, 在所有非本地模型中, 此类比率是最佳的。 此外, 我们通过两个具体的数字示例, 来对非本地模型实施点整体方法(PIM), 以说明另一侧的四角融合率。</s>