原始BGN方法在平均曲率流中的收敛性
Convergence of the original BGN method for mean curvature flow
- Mathematisches Institut, Universität Tübingen（蒂宾根大学数学研究所）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文证明原始BGN方法在平均曲率流中以$h^2$阶收敛于流形距离，通过后向误差分析构造近似流，实现$h^4$阶超收敛，并确认$\ au=h^2$的时间步长关系。
AI中文摘要：
我们证明了用于曲线平均曲率流的原始Barrett--Garcke--Nürnberg (BGN)方法在时间步长$\ au=h^2$时，以$h^2$阶在流形距离下收敛。关键的新技术是后向误差分析，该方法源于常微分方程的数值分析。通过结合离散速度在空间和时间上的展开，我们构造了一个近似流，其插值满足BGN方法且具有改进的缺陷，阶数为$h^4$，其中切向部分为$h^6$阶。这一构造还识别了BGN方法的切向速度，并表明$\ au=O(h^2)$是$\ au$与$h$之间的正确关系。利用改进的缺陷，我们随后使用演化曲面有限元方法的技术证明了数值解对近似流的$H^1$超收敛性，阶数为$h^4$，从而得出在流形距离下的收敛性。数值实验证实了预测的缺陷阶数。
英文摘要：
We prove that the original Barrett--Garcke--Nürnberg (BGN) method for mean curvature flow of curves converges in the manifold distance with order $h^2$ for the time step $τ=h^2$. The key new technique is backward error analysis, which stems from the numerical analysis of ordinary differential equations. By combining the expansions of the discrete velocity in space and time, we construct an approximation flow whose interpolation satisfies the BGN method with an improved defect, of order $h^4$ with tangential part of order $h^6$. This construction also identifies the tangential velocity of the BGN method and shows that $τ=O(h^2)$ is the correct relation between $τ$ and $h$. With the improved defect, we then prove $H^1$ superconvergence of order $h^4$ of the numerical solution to the approximation flow using techniques of evolving surface finite element methods, which yields the convergence in the manifold distance. Numerical experiments confirm the predicted orders of defects.
