MGM: A meshfree geometric multilevel method for systems arising from elliptic equations on point cloud surfaces

TL;DR

This paper introduces a meshfree multilevel method for efficiently solving elliptic PDEs on point cloud surfaces, combining coarsening, meshfree transfer operators, and multilevel iteration, applicable to various discretizations.

Contribution

The paper presents a novel meshfree geometric multilevel method that works with point cloud surfaces and integrates new transfer operators with standard smoothing techniques.

Findings

MGM effectively solves elliptic PDE systems on point clouds.

The method shows favorable convergence and scalability.

MGM outperforms algebraic multigrid in tested scenarios.

Abstract

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.