A New Approach to the Analysis of Parametric Finite Element Approximations to Mean Curvature Flow

Abstract

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.