Meshfree Methods for PDEs on Surfaces
Authors: Jones Andrew Michael 
Published: March 2023

This dissertation focuses on meshfree methods for solving surface partial differential equations (PDEs). These PDEs arise in many areas of
science and engineering where they are used to model phenomena ranging from atmospheric dynamics on earth to chemical signaling on cell
membranes. Meshfree methods have been shown to be effective for solving surface PDEs and are attractive alternatives to mesh-based methods
such as finite differences/elements since they do not require a mesh and can be used for surfaces represented only by a point cloud. The
dissertation is subdivided into two papers and software.

In the first paper, we examine the performance and accuracy of two popular meshfree methods for surface PDEs:generalized moving least
squares (GMLS) and radial basis function-finite differences (RBF-FD). While these methods are computationally efficient and can give high
orders of accuracy for smooth problems, there are no published works that have systematically compared their benefits and shortcomings. We
perform such a comparison by examining their convergence rates for approximating the surface gradient, divergence, and Laplacian on the sphere
and a torus as the resolution of the discretization increases. We investigate these convergence rates also as the various parameters of the
methods are changed. We also compare the overall efficiencies of the methods in terms of accuracy per computation cost.

The second paper is focused on developing a novel meshfree geometric multilevel (MGM) method for solving linear systems associated with
meshfree discretizations of elliptic PDEs on surfaces represented by point clouds. Multilevel (or multigrid) methods are efficient iterative
methods for solving linear systems that arise in numerical PDEs. The key components for multilevel
methods: \grid" coarsening, restriction/ interpolation operators coarsening, and smoothing. The first three components present challenges for
meshfree methods since there are no grids or mesh structures, only point clouds. To overcome these challenges, we develop a geometric point
cloud coarsening method based on Poisson disk sampling, interpolation/ restriction operators based on RBF-FD, and apply Galerkin projections
to coarsen the operator. We test MGM as a standalone solver and preconditioner for Krylov subspace methods on various test problems using
RBF-FD and GMLS discretizations, and numerically analyze convergence rates, scaling, and efficiency with increasing point cloud resolution.
We finish with several application problems.

We conclude the dissertation with a description of two new software packages. The first one is our MGM framework for solving elliptic surface
PDEs. This package is built in Python and utilizes NumPy and SciPy for the data structures (arrays and sparse matrices), solvers (Krylov
subspace methods, Sparse LU), and C++ for the smoothers and point cloud coarsening. The other package is the RBFToolkit which has a Python
version and a C++ version. The latter uses the performance library Kokkos, which allows for the abstraction of parallelism and data management
for shared memory computing architectures. The code utilizes OpenMP for CPU parallelism and can be extended to GPU architectures.