报告问题 (Title)：Generalized Moving Least-Squares Methods for Solving Vector-Valued PDEs on Manifolds（流形上矢量PDE的广义MLS要领）

报告人 (Speaker)： 蒋诗晓 (上海浚？？？萍即笱)

报告时间 (Time)：2025年4月19日（周六）13:30

报告所在 (Place)：校本部GJ303

约请人(Inviter)：秦晓雪

主理部分：理学院数学系

报告摘要：In this talk, we introduce the Generalized Moving Least-Squares (GMLS) method to solve the vector-valued PDEs on smooth 2D manifolds without boundaries embedded in R^3, identified with randomly sampled point cloud data. The approach 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 an appropriate projection so that the complexity of the method 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 the proposed method on various smooth manifolds.