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

TL;DR

This paper develops a new error analysis framework for high-order discontinuous Galerkin methods solving PDEs on point cloud surfaces, focusing on the Laplace-Beltrami operator, with numerical validation on reconstructed surfaces.

Contribution

It introduces a geometric error analysis framework based on Riemann metric tensor approximation, enabling analysis of DG methods on discontinuous geometries.

Findings

Error estimates for DG methods on point cloud surfaces

Validation through numerical experiments on reconstructed geometries

Framework applicable to high-order PDE discretizations

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.