曲率流的参数化有限元逼近  

PARAMETRIC FINITE ELEMENT APPROXIMATIONS OF CURVATURE FLOWS

作　　者：李步扬 Li Buyang(Department of Applied Mathematics,The Hong Kong Polytechnic University Hong Kong,China)

机构地区：[1]香港理工大学应用数学系,中国

出　　处：《计算数学》2022年第2期145-162,共18页Mathematica Numerica Sinica

基　　金：香港研资局优配研究金PolyU15300920(Research Grants Council of Hong Kong SAR,GRF Project No.PolyU15300920)资助.

摘　　要：许多物理现象可以在数学上描述为受曲率驱动的自由界面运动,例如薄膜和泡沫的演变、晶体生长,等等.这些薄膜和界面的运动常依赖于其表面曲率,从而可以用相应的曲率流来描述,其相关自由界面问题的数值计算和误差分析一直是计算数学领域中的难点.参数化有限元法是曲率流的一类有效计算方法,已经能够成功模拟一些曲面在几类基本的曲率流下的演化过程.本文重点讨论曲率流的参数化有限元逼近,它的产生、发展和当前的一些挑战.Many physical phenomena can be mathematically described by curvature-driven free interface motions,such as the evolution of films and foams,crystal growth,and so on.The motion of these films and interfaces often depends on their surface curvature and therefore can be described by the corresponding curvature flows and geometric evolution equations.The numerical computation and error analysis of the related free interface problems are still challenging problems in the field of computational mathematics.The parametric finite element method is a class of effective computational methods for approximating curvature flows,and it has been successful in simulating the evolution of some basic curvature flows,including mean curvature flow,Willmore flow,surface diffusion,and so on.This article focuses on the parametric finite element approximation of curvature flows-its origin,development and some current challenges.

关 键 词：自由界面 曲率流 非线性 参数化有限元 演化有限元 保几何结构 切向速度 网格均匀化 收敛性 误差估计 

分 类 号：O241.82[理学—计算数学]