Solving Partial Differential Equations on Point Clouds

In this paper we present a general framework for solving partial differential equations on manifolds represented by meshless points, i.e., point clouds, without parametrization or connection information. Our method is based on a local approximation of the manifold as well as functions defined on the manifold, such as using least squares, simultaneously in a local intrinsic coordinate system constructed by local principal component analysis (PCA) using K-nearest neighbors (KNN). Once the local reconstruction is available, differential operators on the manifold can be approximated discretely. The framework extends to manifolds of any dimension. The complexity of our method scales well with the total number of points and the true dimension of the manifold (not the embedded dimension). The numerical algorithms, error analysis, and test examples are presented.