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

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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.

Original languageEnglish
Pages (from-to)1592-1617
Number of pages26
JournalSIAM Journal on Numerical Analysis
Volume59
Issue number3
DOIs
Publication statusE-pub ahead of print - 9 Jun 2021

Keywords

  • Convergence
  • Error estimate
  • Evolving surface
  • Finite element method
  • Mean curvature fl ow

ASJC Scopus subject areas

  • Numerical Analysis
  • Computational Mathematics
  • Applied Mathematics

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