An algorithm is proposed for generalized mean curvature flow of closed two-dimensional surfaces, which include inverse mean curvature flow and powers of mean and inverse mean curvature flow. Error estimates are proved for semidiscretizations and full discretizations 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 nonlinear second-order parabolic evolution equations for the normal velocity and normal vector. A convergence proof is presented in the case of finite elements of polynomial degree at least 2 and backward difference formulae of orders 2 to 5. 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, and complemented by experiments for nonconvex surfaces.

中文翻译：

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闭曲面广义平均曲率流收敛有限元算法

提出了一种用于封闭二维曲面广义平均曲率流的算法，该算法包括反平均曲率流和平均幂次方和反平均曲率流。对广义流的半离散化和全离散化证明了误差估计。这里提出和研究的算法结合了演化的表面有限元，其节点确定离散表面，以及用于时间积分的线性隐式后向差分公式。数值方法基于将表面演化耦合到法向速度和法向矢量的非线性二阶抛物线演化方程的系统。在多项式次数至少为 2 的有限元和 2 到 5 阶的后向差分公式的情况下，给出了收敛证明。误差分析结合了稳定性估计和一致性估计，为计算的表面位置、速度、法向向量、法向速度以及平均曲率产生最优阶 $H^1$-范数误差界限。稳定性分析是在矩阵-向量公式中进行的，并且与几何参数无关，几何参数只进入一致性分析。提出了数值实验来说明收敛结果并报告单调量，例如逆平均曲率流的霍金质量，并辅以非凸表面的实验。稳定性分析是在矩阵-向量公式中进行的，并且与几何参数无关，几何参数只进入一致性分析。提出了数值实验来说明收敛结果并报告单调量，例如逆平均曲率流的霍金质量，并辅以非凸表面的实验。稳定性分析是在矩阵-向量公式中进行的，并且与几何参数无关，几何参数只进入一致性分析。提出了数值实验来说明收敛结果并报告单调量，例如逆平均曲率流的霍金质量，并辅以非凸表面的实验。