AI 中文总结
该研究提出带稳定项的全离散Dziuk方法，结合新超逼近结果，证明闭平面曲线曲线缩短流的BGN型离散切向稳定性，实现最优L²收敛性。
AI 中文摘要
我们针对闭平面曲线的曲线缩短流，提出了带分段线性参数有限元的全离散Dziuk方法的稳定化版本。通过精心设计的稳定项，我们能在抛物尺度τ≃h²下证明Barrett--Garcke--Nürnberg（BGN）型的离散切向稳定性——这是连续层面隐藏的特性。结合线性单元反向平均法向量的新超逼近结果，该Dziuk型离散切向稳定性可得到最优L²收敛性。
英文摘要
We propose a stabilized version of the fully discrete Dziuk's method for the curve-shortening flow of a closed planar curve with piecewise linear parametric finite elements. With a carefully designed stabilization term, we are able to show a surprising discrete tangential stability of the Barrett--Garcke--Nürnberg (BGN) type under the parabolic scaling $τ\simeq h^2$---a feature hidden at the continuous level. Together with a new super-approximation result for the reversely averaged normal vector of linear elements, this Dziuk-type discrete tangential stability yields optimal $L^2$ convergence.
