发表机构
The Chinese University of Hong Kong（香港中文大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明经典BGN格式对曲线缩短流的全离散收敛性，通过分离时间与空间误差及精确恒等式，获得匹配的W^{1,∞}误差界，无需额外稳定化。
AI 中文摘要
我们证明了经典Barrett--Garcke--Nürnberg（BGN）格式对光滑嵌入闭平面曲线曲线缩短流的全离散收敛性。主要障碍在于质量形式仅控制法向运动，而切向运动由曲率方程隐式决定并支配参数化。因此，通常的长度衰减估计不能控制全位置更新的扰动。我们通过时间半离散BGN解分离时间误差和空间误差。一个标量法向预解式以及精确的曲率和长度恒等式给出了一致正则性和一阶时间收敛性。在空间分析中，一个自适应的法向-切向范数给出了线性化更新的近压缩估计。Gauss--Lobatto消去和组装节点法向的精确协方差恒等式产生了阶为$h^{k+1}$的$H^1$单步缺陷，而一个精确差分恒等式控制了非线性余项。对于每个固定的$k\ge1$，包括原始分段线性方法，我们在满足$h\le c\tau^2$的周期拟均匀网格上获得了匹配的$W^{1,\infty}$误差界$C(\tau+h^k)$。所有离散步骤均唯一可解，且数值曲线保持正则和嵌入。据我们所知，这是经典BGN曲线缩短格式在没有额外稳定化情况下的首个全离散收敛结果。
英文摘要
We prove fully discrete convergence of the classical Barrett--Garcke--Nürnberg (BGN) scheme for curve-shortening flow of smooth embedded closed planar curves. The main obstruction is that the mass form controls only normal motion, whereas the tangential motion is determined implicitly by the curvature equation and governs the parametrization. The usual length-decay estimate therefore does not control perturbations of the full position update. We separate temporal and spatial errors through the time-semidiscrete BGN solution. A scalar normal resolvent and exact curvature and length identities yield uniform regularity and first-order time convergence. For the spatial analysis, an adapted normal--tangential norm gives a near-contractive estimate for the linearized update. Gauss--Lobatto cancellations and an exact covariance identity for the assembled nodal normals produce an $H^1$ one-step defect of order $h^{k+1}$, while an exact difference identity controls the nonlinear remainder. For every fixed $k\ge1$, including the original piecewise linear method, we obtain the matched $W^{1,\infty}$ error bound $C(τ+h^k)$ on periodic quasi-uniform meshes with $h\le cτ^2$. All discrete steps are uniquely solvable, and the numerical curves remain regular and embedded. To the best of our knowledge, this is the first fully discrete convergence result for the classical BGN curve-shortening scheme without additional stabilization.
