Mesh-free collocation for surface differential operators

Abstract

We present a mesh-free collocation scheme to discretize intrinsic surface differential operators over surface point clouds with given normal vectors. The method is based on Discretization-Corrected Particle Strength Exchange (DC-PSE), which generalizes finite difference methods to mesh-free point clouds and moving Lagrangian particles. The resulting Surface DC-PSE method is derived from an embedding theorem, but we analytically reduce the operator kernels along the surface normals, resulting in an embedding-free, purely surface-intrinsic computational scheme. We benchmark the scheme by discretizing the Laplace-Beltrami operator on a circle and a sphere, and present convergence results for both explicit and implicit solvers. We then showcase the algorithm on the problem of computing mean curvature of an ellipsoid and of the Stanford Bunny by evaluating the surface divergence of the normal vector field with the proposed Surface DC-PSE method.