MGM: A meshfree geometric multilevel method for systems arising from elliptic equations on point cloud surfaces
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.
Cite
@article{arxiv.2204.06154,
title = {MGM: A meshfree geometric multilevel method for systems arising from elliptic equations on point cloud surfaces},
author = {Grady B. Wright and Andrew M. Jones and Varun Shankar},
journal= {arXiv preprint arXiv:2204.06154},
year = {2022}
}