Atlas-based meshless finite differences for elliptic partial differential equations on manifolds represented as point clouds

We present a systematic investigation of atlas-based radial basis function finite difference (RBFFD) methods for solving elliptic partial differential equations on smooth, compact surfaces represented as point clouds. This approach organizes point cloud data according to coordinate charts and computes differentiation weights collectively within each chart, contrasting with traditional RBFFD methods that construct independent stencils at each point. While the theoretical convergence framework has been established, practical implementation strategies and convergence behavior under different refinement regimes remain unexplored. We address these gaps by formalizing implementation strategies for both intrinsic and extrinsic formulations, including practical algorithms for atlas construction and local parameterization from point cloud data. We conduct convergence studies under two refinement scenarios: simultaneous refinement validates theoretical convergence rates, while the fixed-atlas regime reveals accuracy saturation due to geometric discretization error. We compare different local parameterization approaches and investigate weighted variants to improve method performance. Numerical experiments on various complex surfaces demonstrate that atlas-based RBFFD methods accurately solve elliptic problems without requiring explicit surface parameterizations or mesh connectivity.