Parametric finite element methods have achieved great success in approximating the evolution of surfaces under various different geometric flows, including mean curvature flow, Willmore flow, surface diffusion, and so on. However, the convergence of Dziuk’s parametric finite element method, as well as many other widely used parametric finite element methods for these geometric flows, remains open. In this article, we introduce a new approach and a corresponding new framework for the analysis of parametric finite element approximations to surface evolution under geometric flows, by estimating the projected distance from the numerically computed surface to the exact surface, rather than estimating the distance between particle trajectories of the two surfaces as in the currently available numerical analyses. The new framework can recover some hidden geometric structures in geometric flows, such as the full \(H^1\) parabolicity in mean curvature flow, which is used to prove the convergence of Dziuk’s parametric finite element method with finite elements of degree \(k \ge 3\) for surfaces in the three-dimensional space. The new framework introduced in this article also provides a foundational mathematical tool for analyzing other geometric flows and other parametric finite element methods with artificial tangential motions to improve the mesh quality.

参数有限元方法在近似各种不同几何流下的表面演化方面取得了巨大成功，包括平均曲率流、威尔莫尔流、表面扩散等。然而，Dziuk 的参数有限元方法以及许多其他广泛用于这些几何流的参数有限元方法的收敛性仍然开放。在本文中，我们介绍了一种新方法和相应的新框架，用于分析几何流下表面演化的参数有限元近似，通过估计从数值计算表面到精确表面的投影距离，而不是估计之间的距离当前可用的数值分析中两个表面的粒子轨迹。新框架可以恢复几何流中一些隐藏的几何结构，例如平均曲率流中的完整\(H^1\)抛物线，用于证明Dziuk参数有限元方法与次数有限元\( k \ge 3\)表示三维空间中的表面。本文介绍的新框架还提供了一个基础数学工具，用于分析其他几何流和其他具有人工切向运动的参数化有限元方法，以提高网格质量。