SIAM Journal on Numerical Analysis, Volume 59, Issue 3, Page 1592-1617, January 2021.
Dziuk's surface finite element method (FEM) for mean curvature flow has had a significant impact on the development of parametric and evolving surface FEMs for surface evolution equations and curvature flows. However, the convergence of Dziuk's surface FEM for mean curvature flow of closed surfaces still remains open since it was proposed in 1990. In this article, we prove convergence of Dziuk's semidiscrete surface FEM with high-order finite elements for mean curvature flow of closed surfaces. The proof utilizes the matrix-vector formulation of evolving surface FEMs and a monotone structure of the nonlinear discrete surface Laplacian proved in this paper.

中文翻译：

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高阶有限元封闭曲面平均曲率流的 Dziuk 半离散有限元法的收敛性

SIAM Journal on Numerical Analysis，第 59 卷，第 3 期，第 1592-1617 页，2021 年 1 月。
Dziuk 用于平均曲率流的表面有限元方法 (FEM) 对用于表面演化方程的参数化和演化表面 FEM 的开发产生了重大影响和曲率流动。然而，自 1990 年提出以来，Dziuk 对封闭曲面平均曲率流的曲面有限元的收敛性仍然是开放的。 在本文中，我们证明了 Dziuk 半离散曲面有限元对封闭曲面平均曲率流的收敛性与高阶有限元. 该证明利用了演化表面 FEM 的矩阵向量公式和本文中证明的非线性离散表面拉普拉斯算子的单调结构。