A Second-Order Nonlocal Approximation for Manifold Poisson Model with Dirichlet Boundary

TL;DR

This paper establishes the well-posedness and second-order convergence of a nonlocal Poisson model on manifolds with Dirichlet boundary, using Poincare inequality and numerical validation via the point integral method.

Contribution

It proves the well-posedness and optimal second-order localization rate of a nonlocal manifold Poisson model, and demonstrates quadratic convergence numerically.

Findings

Well-posedness of the nonlocal model is established.

The model achieves an optimal second-order localization rate.

Numerical examples confirm quadratic convergence.

Abstract

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.