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

2021/01/04 by Yajie Zhang, Zuoqiang Shi, Zhang, Yajie +1
Engineering · Mathematics · #Numerical methods in engineering #Differential Equations and Numerical Methods #Fractional Differential Equations Solutions

paper · pdf · doi:10.48550/arxiv.2101.01016

Abstract

Recently, we constructed a class of nonlocal Poisson model on manifold under Dirichlet boundary with global O(δ²) truncation error to its local counterpart, where δ 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 2δ-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.

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