Parametric finite element approximations of curvature flows

Abstract

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. 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些挑战.