  This paper is delicate to the development of high-order numerical algorithms
for solving partial differential equations on point clouds. We introduce a new
geometric error analysis framework which requires neither global continuity of
surface patches nor exact geometric information, e.g., embedding maps, tangent
spaces. The new framework provides us a fundamental tool to analyze
discontinuous Galerkin (DG) methods for the Laplace-Beltrami operator on point
clouds. It is illustrated using examples of an interior penalty DG method for
solving the Laplace-Beltrami equation and the corresponding eigenvalue problem
with numerical verification.
