An algorithm is proposed for generalized mean curvature flow of closed two-dimensional surfaces, which include inverse mean curvature flow, powers of mean and inverse mean curvature flow, etc. Error estimates are proven for semi- and full discretisations for the generalized flow. The algorithm proposed and studied here combines evolving surface finite elements, whose nodes determine the discrete surface, and linearly implicit backward difference formulae for time integration. The numerical method is based on a system coupling the surface evolution to non-linear second-order parabolic evolution equations for the normal velocity and normal vector. Convergence proofs are presented in the case of finite elements of polynomial degree at least two and backward difference formulae of orders two to five. The error analysis combines stability estimates and consistency estimates to yield optimal-order $H^1$-norm error bounds for the computed surface position, velocity, normal vector, normal velocity, and therefore for the mean curvature. The stability analysis is performed in the matrix--vector formulation, and is independent of geometric arguments, which only enter the consistency analysis. Numerical experiments are presented to illustrate the convergence results, and also to report on monotone quantities, e.g.~Hawking mass for inverse mean curvature flow. Complemented by experiments for non-convex surfaces.
翻译:对于封闭二维表面的普通平均曲线流,包括反平均曲线流、平均和反平均曲线流的力量等等,提出了一种算法。对于一般流动的半和完全分解,错误估计得到证明。此处拟议的和研究的算法结合了变化中的表面限制元素,这些元素的节点决定离散表面,以及时间整合的线性隐含后向偏差公式。数字方法基于将表面进化与非线性二线性二线性双向偏向演进方程式相结合的系统,正常速度和正常矢量。在多向水平的有限元素中至少提供了两个和后向差异公式的一致证明。错误分析结合了稳定性估计和一致性估计,以产生计算表性表面位置、速度、正常矢量、正常速度和平均曲度的最佳偏差界限。稳定性分析是在正常速度和正常矢量的矩阵-矢量配制中进行,并且独立于几何度参数参数的对比,这些参数只能进入二至二至五级的多向偏差公式的公式公式公式的限定要素。 数值分析将稳定性估算结果与正态实验结果显示到正态的数值。