The use of point sets instead of meshes is becoming increasingly more popular. We present a novel meshless approach for point set processing via partial differential equations (PDEs), which introduces the meshless local Petrol–Galerkin (MLPG) method to the field of graphics. The proposed approach neither needs to construct local or global triangular meshes, nor needs global parameterization. It is only based on local tangent spaces and local interpolated surfaces. By constructing the local symmetric weak form (LSWF) for every point, we can easily assemble PDE-specific mass and stiffness matrices. The corresponding sparse linear system can be solved with an iterative solver. The obtained results showed that the proposed approach can smooth noises on point set surfaces while preserving geometric features. Its efficiency is comparable with the traditional finite element method (FEM). The multiscale model of a point set surface can also be constructed using the proposed approach with different iteration times.