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

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.

arXiv commentto appear in Science China Mathematics

References in corpus (3)

• Analysis of the discontinuous Galerkin method for elliptic problems on surfaces
• Surface Crouzeix-Raviart element for the Laplace-Beltrami equation
• Superconvergence of differential structure for finite element methods on perturbed surface meshes