An intrinsic formulation for radial basis function finite difference methods of partial differential equations on manifolds

This project aims to develop a class of meshless algorithm for solving partial differential equations (PDEs) on surfaces. These equations are used in modeling physical, simulating flow in cell membrane biology, creating computer graphics, and helping engineers to process data from 3D scanners. Due to the curved nature of surfaces, it is one of the fronts for meshless algorithms, i.e., ones that do not require data connectivity to run, to excel. In the past few years, meshless methods for surface PDEs are under active development. There are different varieties, say local vs global methods, and they are all working on extrinsic coordinates. The first part of this project focuses on algorithms development; we will fill in the gap by developing some local methods, namely least- squares (LS) radial basis function finite difference (RBF-FD) methods, that works on intrinsic formulations. All algorithms developed in the project enjoy higher flexibility to work with different metric tensors instead of just the one defined by the surface. The second phase emphasizes on applications including anisotropic diffusion (for MRI processing), pattern formations (for computer graphics), evolving surface (for point cloud processing) problems on surfaces specified by point clouds.