A Second-Order Nonlocal Approximation to Manifold Poisson Models with Neumann Boundary

TL;DR

This work introduces a second-order nonlocal approximation for the Poisson equation on smooth manifolds with homogeneous Neumann boundary, embedding the manifold in a high-dimensional Euclidean space. By expressing the second-order normal derivative via the difference between interior and boundary Laplace–Beltrami operators and augmenting with a carefully constructed boundary term, the authors achieve optimal $O(\delta^2)$ localization rates in $H^1$ for $f\in H^3(\mathcal{M})$, extending to $d\ge 3$ dimensions. The paper establishes well-posedness via a Lax–Milgram framework, analyzes the vanishing-nonlocality limit, and generalizes the framework to include a manifold Poisson model with a potential term, non-homogeneous boundary data, and nonlinearities, all while maintaining energy coercivity. Numerical experiments on $S^2$ and $S^3$ hemispheres corroborate the theoretical convergence rates and demonstrate the method’s compatibility with meshless discretizations such as the Point Integral Method, highlighting practical efficiency in high-dimensional manifold settings.

Abstract

In this paper, we propose a class of nonlocal models to approximate the Poisson model on manifolds with homogeneous Neumann boundary condition, where the manifolds are assumed to be embedded in high dimensional Euclid spaces. In comparison to the existing nonlocal approximation of Poisson models with Neumann boundary, we optimize the truncation error of model by adding an augmented function involving the second order normal derivative along the $2δ$ layer of boundary, with $2δ$ be the nonlocal interaction horizon. The 2nd normal derivative is expressed as the difference between the interior Laplacian and the boundary Laplacian. The concentration of our paper is on the construction of nonlocal model, the well-posedness of model, and its second-order convergence rate to its local counterpart. The localization rate of our nonlocal model is currently optimal among all related works even for the case of high dimensional Euclid spaces.

Figures (2)

• Figure 1: Numerical Example \ref{['hmsphere']}
• Figure 2: Example \ref{['ball']}

Theorems & Definitions (6)

• proof : Proof of Lemma \ref{['Lax']}
• proof
• proof : Proof of Lemma \ref{['lmm2']}
• proof : Proof of Lemma \ref{['lmm5']}
• proof : Proof of Lemma \ref{['lmm3']}
• proof