Generalized Moving Least-Squares Methods for Solving Vector-valued PDEs on Unknown Manifolds

Abstract

In this paper, we extend the Generalized Moving Least-Squares (GMLS) method in two different ways to solve the vector-valued PDEs on unknown smooth 2D manifolds without boundaries embedded in $\mathbb{R}^{3}$, identified with randomly sampled point cloud data. The two approaches are referred to as the intrinsic method and the extrinsic method. For the intrinsic method which relies on local approximations of metric tensors, we simplify the formula of Laplacians and covariant derivatives acting on vector fields at the base point by calculating them in a local Monge coordinate system. On the other hand, the extrinsic method formulates tangential derivatives on a submanifold as the projection of the directional derivative in the ambient Euclidean space onto the tangent space of the submanifold. One challenge of this method is that the discretization of vector Laplacians yields a matrix whose size relies on the ambient dimension. To overcome this issue, we reduce the dimension of vector Laplacian matrices by employing a coordinate transformation. The complexity of both methods scales well with the dimension of manifolds rather than the ambient dimension. We also present supporting numerical examples, including eigenvalue problems, linear Poisson equations, and nonlinear Burgers' equations, to examine the numerical accuracy of proposed methods on various smooth manifolds.