We propose a structure-preserving parametric finite element method (SP-PFEM) for discretizing the surface diffusion of a closed curve in two dimensions (2D) or surface in three dimensions (3D). Here the "structure-preserving" refers to preserving the two fundamental geometric structures of the surface diffusion flow: (i) the conservation of the area/volume enclosed by the closed curve/surface, and (ii) the decrease of the perimeter/total surface area of the curve/surface. For simplicity of notations, we begin with the surface diffusion of a closed curve in 2D and present a weak (variational) formulation of the governing equation. Then we discretize the variational formulation by using the backward Euler method in time and piecewise linear parametric finite elements in space, with a proper approximation of the unit normal vector by using the information of the curves at the current and next time step. The constructed numerical method is shown to preserve the two geometric structures and also enjoys the good property of asymptotic equal mesh distribution. The proposed SP-PFEM is "weakly" implicit (or almost semi-implicit) and the nonlinear system at each time step can be solved very efficiently and accurately by the Newton's iterative method. The SP-PFEM is then extended to discretize the surface diffusion of a closed surface in 3D. Extensive numerical results, including convergence tests, structure-preserving property and asymptotic equal mesh distribution, are reported to demonstrate the accuracy and efficiency of the proposed SP-PFEM for simulating surface diffusion in 2D and 3D.
翻译:我们建议一种结构保全参数性要素法(SP-PFEM),用于将封闭曲线的表面扩散分为两个维度(2D)或三个维度(3D)的表面。这里,“结构保全”是指保护表面扩散流的两个基本几何结构:(一) 保护封闭曲线/地表所包涵的区域/体,和(二) 减少曲线/地表的周边/总表面面积。为简便度,我们首先从封闭曲线在2D中的表面扩散开始,提出一个较弱(变式)的治理方程配方。然后,我们通过使用后向Euler方法在时间和空间使用平偏直线线线线线线线线的线性限定要素使变异的配方分离,同时利用封闭曲线/地表/地表的曲线信息使单位正常矢量适当接近。为了保存两个地球测量结构,并享受无色度等量分布的好特性。拟议的SP-FEMMM是“湿度”隐含(或者几乎半D)的传播过程。