Communications in Mathematical Sciences

Volume 15 (2017)

Number 6

Enforce the Dirichlet boundary condition by volume constraint in point integral method

Pages: 1743 – 1769



Zuoqiang Shi (Yau Mathematical Sciences Center, Tsinghua University, Beijing, China)


Poisson equation on point cloud with Dirichlet boundary condition plays important role in many problems. In this paper, we use the volume constraint proposed by Du et al to handle the Dirichlet boundary condition in the point integral method for Poisson equation on point cloud. We prove that the solution given by volume constraint converges to the true solution as the point cloud converges to the underlying smooth manifold.


point integral method, volume constraint, Dirichlet boundary, Laplace–Beltrami operator, Poisson equation

2010 Mathematics Subject Classification


This research was supported by NSFC Grant 11371220 and 11671005.

Received 23 June 2015

Published 27 June 2017