TY - JOUR
T1 - Point Integral Method for Solving Poisson-Type Equations on Manifolds from Point Clouds with Convergence Guarantees
AU - Li , Zhen
AU - Shi , Zuoqiang
AU - Sun , Jian
JO - Communications in Computational Physics
VL - 1
SP - 228
EP - 258
PY - 2019
DA - 2019/10
SN - 22
DO - http://doi.org/10.4208/cicp.111015.250716a
UR - https://global-sci.org/intro/article_detail/cicp/13354.html
KW - Point integral method, point cloud, Laplace-Beltrami operator, convergence.
AB - <p style="text-align: justify;">Partial differential equations (PDE) on manifolds arise in many areas, including mathematics and many applied fields. Due to the complicated geometrical
structure of the manifold, it is difficult to get efficient numerical method to solve PDE
on manifold. In the paper, we propose a method called point integral method (PIM) to
solve the Poisson-type equations from point clouds. Among different kinds of PDEs,
the Poisson-type equations including the standard Poisson equation and the related
eigenproblem 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-type
equations 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 the
paper, we explain the derivation of the integral equations, describe the point integral
method and its implementation, and present the numerical experiments to demonstrate the convergence of PIM.</p>