Point Integral Method for Solving Poisson-Type Equations on Manifolds from Point Clouds with Convergence Guarantees (Communications in Computational Physics, 2019)
Partial differential equations (PDE) on manifolds arise in many areas, including mathematics and many applied fields. Due to the complicated geometricalstructure of the manifold, it is difficult to get efficient numerical method to solve PDEon manifold. In the paper, we propose a method called point integral method (PIM) tosolve the Poisson-type equations from point clouds. Among different kinds of PDEs,the Poisson-type equations including the standard Poisson equation and the relatedeigenproblem of the Laplace-Beltrami operator are one of the most important. In PIM,the key idea is to derive the integral equations which approximates the Poisson-typeequations and contains no derivatives but only the values of the unknown function.This feature makes the integral equation easy to be discretized from point cloud. In thepaper, we explain the derivation of the integral equations, describe the point integralmethod and its implementation, and present the numerical experiments to demonstrate the convergence of PIM.