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

Partial differential equations on manifolds have been widely studied and plays a crucial role in many subjects. In our previous work, a class of nonlocal models was introduced to approximate the Poisson equation on manifolds that embedded in high dimensional Euclid spaces with Dirichlet and Neumann boundaries. In this paper, we improve the accuracy of such model under Dirichlet boundary by adding a higher order term along a layer adjacent to the boundary. Such term is explicitly expressed by the normal derivative of solution and the mean curvature of the boundary, while the normal derivative is regarded as a variable. All the truncation errors that involve or do not involve such term have been re-analyzed and been significantly reduced. Our concentration is on the well-posedness analysis of the weak formulation corresponding to the nonlocal model and the convergence analysis to its PDE counterpart. The main result of our work is that, such manifold nonlocal model converges to the local Poisson problem in a rate of \mathcal{O}(\delta^2) in H^1 norm, where {\delta} is the parameter that denotes the range of support for the kernel of the nonlocal operators. Such convergence rate is currently optimal among all the nonlocal models according to the literature. Two numerical experiments are included to illustrate our convergence results on the other side.