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Discontinuous Galerkin methods for the Laplace-Beltrami operator on point cloud

Guozhi Dong, Hailong Guo, Zuoqiang Shi

TL;DR

This work develops a high-order discontinuous Galerkin method for PDEs on surfaces represented by point clouds, introducing a novel geometric-error framework that measures approximation quality through the Riemannian metric tensor. By reconstructing patchwise manifolds and using intrinsic parametrizations, the authors derive rigorous bounds that relate metric approximation error to DG convergence for the Laplace-Beltrami operator and its eigenvalue problem. They establish $L^2$ and intermediate norms error estimates that depend on the polynomial degrees of geometry and solution spaces, and prove eigenvalue convergence within the Babuška-Osborn framework, including conditions for possible superconvergence. Numerical experiments on unit spheres and tori from point clouds validate the theoretical rates and reveal geometric superconvergence phenomena for certain even-degree patches, illustrating the practical impact of the metric-based analysis on meshfree, high-order surface PDE solvers.

Abstract

This paper is dedicated to the development of numerical analysis for high-order methods solving partial differential equations on scattered point clouds. We build a novel geometric error analysis framework by estimating the error in the approximation of the Riemann metric tensor. The innovative framework serves as a fundamental tool for analyzing discontinuous Galerkin methods applied to the Laplace-Beltrami operator on possibly discontinuous geometry. We provide numerical examples on patchy surfaces reconstructed from point clouds to support our theoretical findings.

Discontinuous Galerkin methods for the Laplace-Beltrami operator on point cloud

TL;DR

This work develops a high-order discontinuous Galerkin method for PDEs on surfaces represented by point clouds, introducing a novel geometric-error framework that measures approximation quality through the Riemannian metric tensor. By reconstructing patchwise manifolds and using intrinsic parametrizations, the authors derive rigorous bounds that relate metric approximation error to DG convergence for the Laplace-Beltrami operator and its eigenvalue problem. They establish and intermediate norms error estimates that depend on the polynomial degrees of geometry and solution spaces, and prove eigenvalue convergence within the Babuška-Osborn framework, including conditions for possible superconvergence. Numerical experiments on unit spheres and tori from point clouds validate the theoretical rates and reveal geometric superconvergence phenomena for certain even-degree patches, illustrating the practical impact of the metric-based analysis on meshfree, high-order surface PDE solvers.

Abstract

This paper is dedicated to the development of numerical analysis for high-order methods solving partial differential equations on scattered point clouds. We build a novel geometric error analysis framework by estimating the error in the approximation of the Riemann metric tensor. The innovative framework serves as a fundamental tool for analyzing discontinuous Galerkin methods applied to the Laplace-Beltrami operator on possibly discontinuous geometry. We provide numerical examples on patchy surfaces reconstructed from point clouds to support our theoretical findings.

Paper Structure

This paper contains 18 sections, 13 theorems, 108 equations, 4 figures, 5 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Patches reconstructed from point clouds and their error analysis
  3. Geometry and its patchwise approximation
  4. Reference mesh on point clouds
  5. Patch reconstruction from point cloud
  6. Geometric error analysis
  7. DG methods in curved domain and their error analysis
  8. Function spaces associated to DG methods
  9. DG methods for PDEs on curved domain
  10. Error analysis for Laplace-Beltrami equation
  11. Error analysis for the eigenvalue problem
  12. Numerical results
  13. Geometric error
  14. PDEs on patches reconstructed from a point cloud of the unit sphere
  15. PDEs on patches reconstructed from a point cloud of a torus
  16. ...and 3 more sections

Key Result

Proposition 2.7

Suppose the Newton iteration in Algorithm alg:point_estimate stops with a sufficiently small error. Then the reconstructed nodal point $\{\psi^i=\pi_h^k(x^i)\}_{i\in\mathbb N} \subset \hat{\Gamma}_h^{k,j}$ and the precise nodal points $\{\xi^i=\pi(x^i)\}_{i\in\mathbb N}\subset \Gamma$ satisfy the fo In particular, where $\psi^i_{\Omega_h}$ and $\xi^i_{\Omega_h}$ are the nodal points of $\psi^i$ a

Figures (4)

  • Figure 1: (Color online) Initial meshes constructed from point cloud. Left: Point cloud; Middle: Reference mesh reconstructed by SPR SPR; Right: Zoom-in of the point cloud and triangular meshes.
  • Figure 2: (Color online) Numerical results of Laplace-Beltrami equation on the unit sphere. Left (a): $L_2$ error; Right (b): $H_1$ error.
  • Figure 3: (Color online) Numerical results of Laplace-Beltrami equation on torus point could. Left (a): $L_2$ error; Right (b): $H_1$ error.
  • Figure 4: (Color online) Numerical results of Laplace-Beltrami eigenvalue problem on torus point cloud. Left (a): $k=l=2$; Right (b): $k=l=4$.

Theorems & Definitions (30)

  • Definition 2.1: $k$-th order patchwise manifold
  • Definition 2.2
  • Remark 2.3
  • Remark 2.4
  • Definition 2.5
  • Proposition 2.7
  • Lemma 2.8
  • proof : Proof of Proposition \ref{['prop:nodal_accu']}
  • Remark 2.9
  • Lemma 2.10
  • ...and 20 more