Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A convergent finite element method with minimal deformation rate for mean curvature flow

Tiantian Huang, Buyang Li, Rong Tang

Abstract

We propose and analyze a fully discrete parametric finite element method with minimal deformation rate (MDR) for simulating the mean curvature flow of general closed surfaces in three dimensions. The method is formulated from a coupled system that enforces the mean curvature flow law for the normal velocity while introducing an artificial tangential velocity that minimizes the deformation-rate energy, thereby preserving mesh quality without requiring remeshing or reparametrization. An $L^{2}$-projected averaged normal vector is used in the scheme to facilitate a rigorous convergence analysis. Within the projected--distance framework, we establish the first complete convergence proof for a parametric finite element method that incorporates the MDR tangential motion without relying on evolution equations for the mean curvature or the normal vector, achieving optimal-order error estimates for finite elements of degree $k \ge 3$. Numerical experiments corroborate the theoretical results and demonstrate that the proposed MDR method maintains mesh quality comparable to the Barrett--Garcke--Nürnberg method, for which convergence has not yet been established.

A convergent finite element method with minimal deformation rate for mean curvature flow

Abstract

We propose and analyze a fully discrete parametric finite element method with minimal deformation rate (MDR) for simulating the mean curvature flow of general closed surfaces in three dimensions. The method is formulated from a coupled system that enforces the mean curvature flow law for the normal velocity while introducing an artificial tangential velocity that minimizes the deformation-rate energy, thereby preserving mesh quality without requiring remeshing or reparametrization. An -projected averaged normal vector is used in the scheme to facilitate a rigorous convergence analysis. Within the projected--distance framework, we establish the first complete convergence proof for a parametric finite element method that incorporates the MDR tangential motion without relying on evolution equations for the mean curvature or the normal vector, achieving optimal-order error estimates for finite elements of degree . Numerical experiments corroborate the theoretical results and demonstrate that the proposed MDR method maintains mesh quality comparable to the Barrett--Garcke--Nürnberg method, for which convergence has not yet been established.
Paper Structure (39 sections, 23 theorems, 320 equations, 3 figures)

This paper contains 39 sections, 23 theorems, 320 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Basic settings and main results
  3. Distance projection onto the exact surface
  4. Initial triangulation
  5. The numerical scheme
  6. Distance projection of $\Gamma_h^m$ onto $\Gamma^m$
  7. Notation for finite element functions
  8. Lift and inverse lift
  9. Main theoretical result
  10. The underlying framework
  11. Notaions
  12. Approximation properties of the interpolated surface ${\hat{\Gamma}^m_{h, *}}$
  13. Induction assumptions
  14. Estimates of the averaged normal vectors
  15. Poincaré Inequalities for Vector Functions
  16. ...and 24 more sections

Key Result

Theorem 2.1

Suppose that the flow map $\phi:\Gamma^0\times[0,T]\rightarrow {\mathbb R}^3$ of the mean curvature flow and its inverse map $\phi(\cdot,t)^{-1}:\Gamma(t)\rightarrow\Gamma^0$ are both sufficiently smooth, uniformly with respect to $t\in[0,T]$, and the initial approximation $\Gamma_h^0$ is sufficient

Figures (3)

  • Figure 6.1: Errors and convergence rates (Example \ref{['Example1']}).
  • Figure 6.2: Mean curvature flow for Dumbbell surface in Example \ref{['Example2']}
  • Figure 6.3: Mean curvature flow for rectangular box in Example \ref{['Example3']}

Theorems & Definitions (42)

  • Theorem 2.1: Convergence of the MDR method
  • Remark 3.1
  • Lemma 3.2: kovacs2017convergence
  • Lemma 3.3
  • Lemma 3.4
  • proof
  • Lemma 3.5
  • Lemma 3.6
  • Remark 3.7
  • Lemma 3.8: Super-approximation estimates for product of functions
  • ...and 32 more