  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.
