A Second-Order Nonlocal Approximation for Surface Poisson Model with Dirichlet Boundary
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Partial differential equations on manifolds have been widely studied and plays a crucial role in many subjects. In our previous work, a class of integral equations was introduced to approximate the Poisson problems on manifolds with Dirichlet and Neumann type boundary conditions. In this paper, we restrict our domain into a compact, two dimensional manifold(surface) embedded in high dimensional Euclid space with Dirichlet boundary. Under such special case, a class of more accurate nonlocal models are set up to approximate the Poisson model. One innovation of our model is that, the normal derivative on the boundary is regarded as a variable so that the second order normal derivative can be explicitly expressed by such variable and the curvature of the boundary. Our concentration is on the well-posedness analysis of the weak formulation corresponding to the integral model and the study of convergence to its PDE counterpart. The main result of our work is that, such surface nonlocal model converges to the standard Poisson problem in a rate of 𝒪(δ^2) in H^1 norm, where δ is the parameter that denotes the range of support for the kernel of the integral operators. Such convergence rate is currently optimal among all integral models according to the literature. Two numerical experiments are included to illustrate our convergence analysis on the other side.
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