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<p>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 of patchy surfaces reconstructed from point clouds to support our theoretical findings.</p>

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

  • 期刊论文

  • 发表日期：

    2025年3月21日

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  • Guozhi Dong,
  • Hailong Guo,
  • Zuoqiang Shi

摘要

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 of patchy surfaces reconstructed from point clouds to support our theoretical findings.