SIAM Journal on Numerical Analysis, Volume 62, Issue 5, Page 2172-2195, October 2024.
Abstract. A novel evolving surface finite element method, based on a novel equivalent formulation of the continuous problem, is proposed for computing the evolution of a closed hypersurface moving under a prescribed velocity field in two- and three-dimensional spaces. The method improves the mesh quality of the approximate surface by minimizing the rate of deformation using an artificial tangential motion. The transport evolution equations of the normal vector and the extrinsic Weingarten matrix are derived and coupled with the surface evolution equations to ensure stability and convergence of the numerical approximations. Optimal-order convergence of the semidiscrete evolving surface finite element method is proved for finite elements of degree [math]. Numerical examples are provided to illustrate the convergence of the proposed method and its effectiveness in improving mesh quality on the approximate evolving surface.

中文翻译：

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规定速度场下表面演化的人工切向运动收敛演化有限元方法


《SIAM 数值分析杂志》，第 62 卷，第 5 期，第 2172-2195 页，2024 年 10 月。

抽象的。提出了一种基于连续问题的新颖等效公式的新型演化表面有限元方法，用于计算二维和三维空间中指定速度场下运动的闭合超曲面的演化。该方法通过使用人工切向运动最小化变形率来提高近似表面的网格质量。推导了法向量和外在Weingarten矩阵的输运演化方程，并将其与表面演化方程耦合，以确保数值近似的稳定性和收敛性。对于度数有限元，证明了半离散演化面有限元法的最优阶收敛性。提供了数值例子来说明该方法的收敛性及其在提高近似演化曲面上网格质量方面的有效性。