  We develop a new meshfree geometric multilevel (MGM) method for solving
linear systems that arise from discretizing elliptic PDEs on surfaces
represented by point clouds. The method uses a Poisson disk sampling-type
technique for coarsening the point clouds and new meshfree
restriction/interpolation operators based on polyharmonic splines for
transferring information between the coarsened point clouds. These are then
combined with standard smoothing and operator coarsening methods in a V-cycle
iteration. MGM is applicable to discretizations of elliptic PDEs based on
various localized meshfree methods, including RBF finite differences (RBF-FD)
and generalized finite differences (GFD). We test MGM both as a standalone
solver and preconditioner for Krylov subspace methods on several test problems
using RBF-FD and GFD, and numerically analyze convergence rates, efficiency,
and scaling with increasing point cloud sizes. We also perform a side-by-side
comparison to algebraic multigrid (AMG) methods for solving the same systems.
Finally, we further demonstrate the effectiveness of MGM by applying it to
three challenging applications on complicated surfaces: pattern formation,
surface harmonics, and geodesic distance.
