| As a way of data representation,point cloud is widely used in data mining,machine learning and other fields.From the perspective of mathematics and calculation,we can obtain some information of the point cloud from the properties of the solutions of some partial differential equations on itself.Therefore,the numerical solution of partial differential equations on the point cloud is of great significance.Due to the complexity of global mesh generation,the methods of solving partial differential equation on point cloud based on mesh generation have low computational efficiency,while the existing methods of solving partial differential equations on point cloud not based on mesh generation have low precision.The traditional meshless methods are only suitable for solving partial differential equations in a region of Euclidean space.How to design an efficient and high-precision algorithm for solving partial differential equations on point cloud is an important research topic.This paper is devoted to the study of an efficient and highprecision algorithm for solving partial differential equations on point cloud.The main results of this paper are as follows:1.By equivalently extend the function on the manifold along the normal direction to the embedded space and use the moving least squares approximation in the embedded space,we extend the traditional meshless Galerkin method to the manifold and construct a new function approximation space.We verify its effectiveness through a series of numerical experiments.Compared with the existing meshless methods,this method can achieve higher accuracy.2.For the proposed method,we design a weight function of the corresponding moving least squares so that the integrand functions in the scheme have better smoothness,so we can compute the numerical integration more efficient. |
