平均曲率流的动态Ritz投影及参数有限元方法的最优${\bf L^2}$收敛性
Dynamic Ritz projection of mean curvature flow and optimal ${\bf L^2}$ convergence of parametric FEM
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中文总结 AI 辅助
本文提出动态Ritz投影方法，用于分析三维闭曲面平均曲率流参数有限元逼近，证明在$L^\infty(0,T;L^2)$范数下的最优阶收敛性，包括分片线性元情形。
中文摘要 AI 辅助
本文发展了一种新方法，用于研究三维空间中闭曲面平均曲率流的参数有限元逼近的收敛性。在该方法中，误差分析通过将数值解与本文引入的平均曲率流的动态Ritz投影进行比较来进行，而非像文献中常用做法那样与平均曲率流的插值进行比较。建立了动态Ritz投影在逼近平均曲率流时在$L^2$和$W^{1,p}$范数下的误差。利用这些结果，证明了闭曲面平均曲率流的参数有限元方法在$L^\infty(0,T;L^2)$范数下的最优阶收敛性，包括使用分片线性有限元的参数有限元方法的收敛性。
英文摘要
A new approach is developed to study the convergence of parametric finite element approximations to the mean curvature flow of closed surfaces in three-dimensional space. In this approach, the error analysis is conducted by comparing the numerical solution to a dynamic Ritz projection of the mean curvature flow introduced in this paper, rather than an interpolation of the mean curvature flow, as commonly used in the literature. The errors associated with the dynamic Ritz projection in approximating the mean curvature flow are established in the $L^2$ and $W^{1,p}$ norms. Leveraging these results, optimal-order convergence of parametric finite element methods for mean curvature flow of closed surfaces in the $L^\infty(0,T;L^2)$ norm is proved, including the convergence of parametric finite element methods with piecewise linear finite elements.
发表机构
- The Hong Kong Polytechnic University（香港理工大学）
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