AI 中文总结
研究封闭曲面平均曲率流，提出用拉格朗日乘子保持能量递减结构的有限元方法，基于Huisken方程，用BDF和隐式亚当斯方法离散时间，证明相关收敛性并推导误差界，数值实验验证有效性及低计算开销。
AI 中文摘要
我们提出并分析了一类用于封闭曲面平均曲率流的演化曲面有限元方法，使用拉格朗日乘子来保持能量递减结构。该算法基于由Huisken法向矢量和平均曲率演化方程驱动的解公式。时间离散化对抛物几何变量使用线性隐式向后差分公式（BDF），对节点位置使用隐式亚当斯更新。该方法还考虑了最小变形率类型的人工切向速度。得到的全离散算法在每个时间步都减小面积，其衰减率由计算出的平均曲率确定。我们证明了离散拉格朗日乘子的局部存在性和唯一性，以及在弱正则性假设下简化牛顿迭代计算的收敛性。在更强的正则性假设下，我们推导了对于多项式次数\(k\geq2\)的有限元以及\(q\)步BDF和\(q\)步隐式亚当斯方法（\(2\leq q\leq5\)），在有和没有最小变形率切向运动情况下，\(H^1\)范数下\(h^k+\tau^q\)阶的最优误差界。球体平均曲率流的数值实验证实了预测的收敛速率，并表明拉格朗日乘子校正仅带来很小的计算开销，且基本与网格大小和时间步长无关。
英文摘要
We propose and analyze a class of evolving surface finite element methods for mean curvature flow of closed surfaces using Lagrange multipliers for preserving the energy-decreasing structure. The algorithm is based on the solution-driven formulation using Huisken's evolution equations for the normal vector and the mean curvature. The time discretizations use linearly implicit backward difference formulas (BDF) for the parabolic geometric variables and implicit Adams updates for the nodal positions. The approach also accommodates artificial tangential velocities of minimal-deformation-rate type. The resulting fully discrete algorithms are area-decreasing at every time step, with a prescribed decay rate determined by the computed mean curvature. We prove local existence and uniqueness of the discrete Lagrange multiplier and convergence of a simplified Newton iteration for its computation under weak regularity assumptions. Under stronger regularity assumptions, as used in the convergence theory for the underlying evolving surface finite element method, we derive optimal-order error bounds of order $h^k+τ^q$ in the $H^1$-norm for finite elements of polynomial degree $k\ge 2$ and $q$-step BDF and $q$-step implicit Adams methods with $2\le q\le 5$, both without and with minimal-deformation-rate tangential motion. Numerical experiments for mean curvature flow of a sphere confirm the predicted convergence rates and show that the Lagrange-multiplier correction entails only a small computational overhead that is essentially independent of the mesh size and the time step size.
