We present a meshfree collocation scheme to discretize intrinsic surface differential operators over scalar fields on smooth curved surfaces with given normal vectors and a non-intersecting tubular neighborhood. The method is based on Discretization-Corrected Particle Strength Exchange (DC-PSE), which generalizes finite difference methods to meshfree point clouds. The proposed Surface DC-PSE method is derived from an embedding theorem, but we analytically reduce the operator kernels along surface normals to obtain a purely intrinsic computational scheme over surface point clouds. We benchmark Surface DC-PSE by discretizing the Laplace-Beltrami operator on a circle and a sphere, and we present convergence results for both explicit and implicit solvers. We then showcase the algorithm on the problem of computing Gauss and mean curvature of an ellipsoid and of the Stanford Bunny by approximating the intrinsic divergence of the normal vector field. Finally, we compare Surface DC-PSE with Surface Finite Elements (SFEM) and Diffuse-Interface Finite Elements (DI FEM) in a validation case.

翻译：我们提出了一种无网格配置方案，用于在给定法向量且具有非相交管状邻域的光滑曲面上，离散化标量场的本征表面微分算子。该方法基于离散校正粒子强度交换（DC-PSE），它将有限差分法推广到无网格点云。所提出的表面DC-PSE方法源于一个嵌入定理，但我们通过解析方法沿曲面法线约化算子核，得到了一个纯粹基于曲面点云的本征计算方案。通过在圆和球面上离散化Laplace-Beltrami算子，我们对表面DC-PSE进行了基准测试，并展示了显式与隐式求解器的收敛结果。接着，我们通过近似法向量场的本征散度来计算椭球体和斯坦福兔子的高斯曲率与平均曲率，以此展示该算法。最后，在验证案例中，我们将表面DC-PSE与表面有限元法（SFEM）和扩散界面有限元法（DI FEM）进行了比较。