Convergence of Dziuk's semidiscrete finite element method for mean curvature flow of closed surfaces with high-order finite elements

Abstract

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.