Speaker: Grady Wright (Boise State University)

Title: A meshfree geometric multilevel method for elliptic PDEs on surfaces

Abstract: Multilevel/multigrid methods can be efficient and robust solvers and preconditioners for linear systems that arise from discretizing elliptic partial differential equations (PDEs). Geometric multilevel methods typically use grid-based restriction and interpolation operators on a hierarchy of coarser grids to efficiently solve these systems. In this paper, we remove the use of grids by developing new meshfree (or meshless) restriction/interpolation operators based on radial basis functions (RBFs) and combine these with a meshfree coarsening strategy. We apply these methods as a solver (and preconditioner) in a V-cycle iteration to meshfree discretizations of elliptic equations on surfaces represented by point clouds. The new method is applicable to various localized meshfree methods, including RBF finite differences (RBF-FD) and generalized finite differences (GFD). We present results on efficiency and convergence rates of the new meshfree multilevel method and demonstrate its improved performance over algebraic multigrid (AMG).  We also use the method in three challenging applications to further demonstrate its effectiveness: computing surface harmonics, pattern formation, and geodesic distance.