标题： 具有Dirichlet边值条件的曲面平均曲率流的一个收敛算法 显示英文标题

标题： A convergent algorithm for mean curvature flow of surfaces with Dirichlet boundary conditions

摘要： 我们针对具有固定边界的曲面的平均曲率流（MCF）建立了空间半离散化的收敛性结果。我们的分析基于Huisken关于平均曲率和法向量的演化方程，这使得我们可以精确控制离散化误差，并得到度数为$p \geq 2$的分段多项式离散空间下的最优误差估计。借鉴Kovács、Li、Lubich及其合作者最近为闭合曲面开发的技术，我们将这些思想扩展到带边界的曲面，通过为平均曲率和法向量制定适当的边界条件来实现这一点。这些边界处理对于证明收敛性至关重要。我们的分析核心是一种经典的误差分裂策略，使用辅助离散函数来近似曲面几何、平均曲率和法向量。我们对每个变量估计两种类型的误差，以严格评估稳定性和一致性。为了有效处理法向量的边界条件，我们在分析中引入了非线性Ritz投影。由此，我们得到了曲面位置、速度、平均曲率和法向量的最优$H^1$误差估计。我们的理论结果通过数值实验得到了验证。 显示英文摘要

摘要： We establish convergence results for a spatial semidiscretization of Mean Curvature Flow (MCF) for surfaces with fixed boundaries. Our analysis is based on Huisken's evolution equations for the mean curvature and the normal vector, enabling precise control of discretization errors and yielding optimal error estimates for discrete spaces with piecewise polynomials of degree $p \geq 2$. Building on techniques recently developed by Kov\'acs, Li, Lubich, and collaborators for closed surfaces, we extend these ideas to surfaces with boundaries by formulating appropriate boundary conditions for both the mean curvature and the normal vector. These boundary treatments are essential for proving convergence. The core of our analysis involves a classical error splitting strategy using auxiliary discrete functions that approximate the surface geometry, the mean curvature, and the normal vector. We estimate two types of errors for each variable to rigorously assess both stability and consistency. To effectively handle boundary conditions for the normal vector, we introduce a nonlinear Ritz projection into the analysis. As a result, we derive optimal $H^1$ error estimates for the surface position, velocity, mean curvature, and normal vector. Our theoretical findings are corroborated by numerical experiments.