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

Guozhi Dong, Hailong Guo, Zuoqiang Shi

A generic geometric error analysis framework for numerical solution of PDEs on sub-manifolds is established. It requires no global continuity on the patches, and also no embedding of the manifolds and tangential projections. This is different from the existing analysis in the literature, where geometry is assumed to be continuous and errors are mostly residuals of polynomial interpolations. It is applied to analyzing discontinuous Galerkin (DG) methods for the Laplace-Beltrami operator on point clouds. The idea is illustrated through particular examples of an interior penalty DG method for solving the Laplace-Beltrami equation and the corresponding eigenvalue problem with numerical verification.

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