MGM: A MESHFREE GEOMETRIC MULTILEVEL METHOD FOR SYSTEMS ARISING FROM ELLIPTIC EQUATIONS ON POINT CLOUD SURFACES

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 mesh -free restriction/interpolation operators based on polyharmonic splines for transferring information between the coarsened point clouds. These are then combined with standard smoothing and op-erator 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 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.

Automated summary

We propose a new meshfree geometric multilevel (MGM) method for solving linear systems arising from discretizing elliptic PDEs on point cloud surfaces. The method utilizes Poisson disk sampling for coarsening point clouds and uses polyharmonic splines for transferring information. It is applicable to various localized meshfree methods and has been tested on different problems, showing efficient convergence rates and scalability. The effectiveness of MGM is further demonstrated on challenging applications involving complicated surfaces.