曲率流的参数化有限元逼近

doi:10.12286/jssx.j2021-0871

计算数学 ›› 2022, Vol. 44 ›› Issue (2): 145-162.DOI: 10.12286/jssx.j2021-0871

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曲率流的参数化有限元逼近

李步扬

香港理工大学应用数学系

收稿日期:2021-09-24 出版日期:2022-05-14 发布日期:2022-05-06
作者简介:李步扬,香港理工大学应用数学系副教授,现为计算数学杂志SIAM Journal on Numerical Analysis和Mathematics of Computation的编委.作者2005年本科毕业于山东大学、2012年在香港城市大学数学系获博士学位,博士毕业后入职南京大学,2015年作为洪堡学者赴University of Tübingen从事研究一年,后于2016年加入香港理工大学工作至今.主要研究领域为偏微分方程的数值计算方法,包括曲率流的数值逼近和误差分析、非线性色散方程不光滑解的计算方法、不可压Navier——Stokes方程的数值分析、非线性抛物方程的L^p范数误差估计、分数阶偏微分方程的高精度求解、热敏电阻方程的适定性和数值分析,等等.
基金资助:
香港研资局优配研究金PolyU15300920(Research Grants Council of Hong Kong SAR, GRF Project No.PolyU15300920)资助.

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李步扬. 曲率流的参数化有限元逼近[J]. 计算数学, 2022, 44(2): 145-162.

Li Buyang. PARAMETRIC FINITE ELEMENT APPROXIMATIONS OF CURVATURE FLOWS[J]. Mathematica Numerica Sinica, 2022, 44(2): 145-162.

PARAMETRIC FINITE ELEMENT APPROXIMATIONS OF CURVATURE FLOWS

Li Buyang

Department of Applied Mathematics, The Hong Kong Polytechnic University Hong Kong, China

Received:2021-09-24 Online:2022-05-14 Published:2022-05-06

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.

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

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.

Key words: free interface, curvature flow, nonlinear, parametric finite element method, evolving finite elements, structure preserving, tangential velocity, mesh points distribution, convergence, error estimation

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曲率流的参数化有限元逼近

PARAMETRIC FINITE ELEMENT APPROXIMATIONS OF CURVATURE FLOWS

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