[Paper Review] Convergence of the Point Integral method for the Poisson equation with Dirichlet boundary on point cloud

Zuoqiang Shi, Jian Sun|arXiv (Cornell University)|Dec 16, 2013

Numerical methods in engineering13 references4 citations

TL;DR

This paper proves the convergence of the Point Integral Method (PIM) for solving the Poisson equation with Dirichlet boundary conditions on point clouds. By approximating the problem via a Robin-type integral formulation and leveraging kernel-based integral operators, the method achieves optimal convergence rates under appropriate mesh and kernel parameter conditions.

ABSTRACT

The Poisson equation on manifolds plays an fundamental role in many applications. Recently, we proposed a novel numerical method called the Point Integral method (PIM) to solve the Poisson equations on manifolds from point clouds. In this paper, we prove the convergence of the point integral method for solving the Poisson equation with the Dirichlet boundary condition.

Motivation & Objective

• To establish the convergence of the Point Integral Method (PIM) for solving the Poisson equation with Dirichlet boundary conditions on point clouds.
• To address the challenge of applying integral approximations to Dirichlet problems, which require normal derivatives not provided in the boundary condition.
• To use a Robin problem as a regularized approximation to the Dirichlet problem, enabling the use of integral formulations.
• To derive error bounds for the discrete PIM solution in terms of mesh size $ h $, kernel parameter $ t $, and boundary regularity.
• To prove that the discrete $ L^2 $ and $ H^1 $ norms of the solution converge to their continuous counterparts under appropriate parameter scaling.

Proposed method

• Approximate the Laplace-Beltrami operator on a manifold using a kernel-based integral operator involving $ R_t(x,y) $ and $ \bar{R}_t(\bx,\by) $, defined via a radial basis function $ R $.
• Transform the Poisson equation into an integral equation by leveraging the identity $ -\int_{\mathcal{M}} \Delta_{\mathcal{M}} u(\by) \bar{R}_t(\bx,\by) d\mu_\by \approx \frac{1}{t} \int_{\mathcal{M}} R_t(\bx,\by)(u(\bx)-u(\by)) d\mu_\by - 2 \int_{\partial\mathcal{M}} \bar{R}_t(\bx,\by) \frac{\partial u}{\partial \bn}(\by) d\tau_\by $.
• Use a Robin-type boundary condition $ u + \beta \frac{\partial u}{\partial \bn} = b $ as a regularized approximation to the Dirichlet problem, with small $ \beta $.
• Discretize the integral equation using a point cloud $ \{\bp_i\} $ with weights $ V_i $, approximating the manifold measure $ \mu $ and boundary measure $ \tau $.
• Define discrete solution vectors $ \bu $ and derive a linear system based on the integral approximation, with kernel weights depending on $ |\bp_i - \bp_j|^2 $.
• Establish error estimates by bounding the difference between continuous and discrete $ L^2 $ and $ H^1 $ norms using kernel decay and mesh regularity.

Experimental results

Research questions

• RQ1 Can the Point Integral Method (PIM) be rigorously proven to converge for the Poisson equation with Dirichlet boundary conditions on point clouds?
• RQ2 How does the approximation error of the PIM depend on the mesh size $ h $, kernel parameter $ t $, and boundary regularity?
• RQ3 What is the optimal scaling between $ h $ and $ t $ that ensures convergence of the discrete solution to the true solution?
• RQ4 Can the Dirichlet problem be effectively approximated using a Robin-type formulation in the context of kernel-based integral methods?
• RQ5 How do the discrete $ L^2 $ and $ H^1 $ norms of the PIM solution relate to the continuous norms of the exact solution?

Key findings

• The discrete $ L^2 $ norm of the PIM solution satisfies $ \left| \|u_{t,h}\|_{L^2(\mathcal{M})}^2 - \sum_{i=1}^n u_i^2 V_i \right| \leq \frac{Ch}{t^{1/2}} \left( \|u_{t,h}\|_{H^1(\mathcal{M})}^2 + t^{3/2} \|f\|_{\infty}^2 \right) $, with $ \frac{h}{t^{1/2}} \leq \frac{1}{2} $.
• The discrete $ L^2 $ norm on the boundary satisfies $ \left| \|u_{t,h}\|_{L^2(\partial\mathcal{M})}^2 - \sum_{l \in I_S} u_l^2 A_l \right| \leq \frac{Ch}{t} \left( \|u_{t,h}\|_{H^1(\mathcal{M})}^2 + t^{3/2} \|f\|_{\infty}^2 \right) $.
• Under the condition $ \frac{h}{t^{1/2}} \leq \frac{1}{2} $, the discrete $ H^1 $-like norm is bounded by $ \left( \sum u_i^2 V_i \right)^{1/2} + t^{1/4} \left( \sum u_l^2 A_l \right)^{1/2} \leq C \|u_{t,h}\|_{H^1(\mathcal{M})} + C \sqrt{h} \, t^{3/4} \|f\|_{\infty} $.
• The method achieves convergence in the $ H^1(\mathcal{M}) $-norm as $ h \to 0 $ and $ t \to 0 $, provided $ \frac{h}{t^{1/2}} \to 0 $, ensuring optimal error decay.
• The error estimates are uniform in the kernel parameter $ t $, and the convergence rate is controlled by the interplay between $ h $, $ t $, and the regularity of the solution.
• The analysis confirms that the PIM is a stable and convergent method for solving the Dirichlet problem on point clouds, even in high-dimensional ambient spaces.