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A convergent algorithm for mean curvature flow of surfaces with Dirichlet boundary conditions

Bárbara Solange Ivaniszyn, Pedro Morin, M. Sebastián Pauletti

Abstract

We establish convergence results for a spatial semidiscretization of Mean Curvature Flow (MCF) for surfaces with fixed boundaries. Our analysis is based on Huisken's evolution equations for the mean curvature and the normal vector, enabling precise control of discretization errors and yielding optimal error estimates for discrete spaces with piecewise polynomials of degree $p \geq 2$. Building on techniques recently developed by Kovács, Li, Lubich, and collaborators for closed surfaces, we extend these ideas to surfaces with boundaries by formulating appropriate boundary conditions for both the mean curvature and the normal vector. These boundary treatments are essential for proving convergence. The core of our analysis involves a classical error splitting strategy using auxiliary discrete functions that approximate the surface geometry, the mean curvature, and the normal vector. We estimate two types of errors for each variable to rigorously assess both stability and consistency. To effectively handle boundary conditions for the normal vector, we introduce a nonlinear Ritz projection into the analysis. As a result, we derive optimal $H^1$ error estimates for the surface position, velocity, mean curvature, and normal vector. Our theoretical findings are corroborated by numerical experiments.

A convergent algorithm for mean curvature flow of surfaces with Dirichlet boundary conditions

Abstract

We establish convergence results for a spatial semidiscretization of Mean Curvature Flow (MCF) for surfaces with fixed boundaries. Our analysis is based on Huisken's evolution equations for the mean curvature and the normal vector, enabling precise control of discretization errors and yielding optimal error estimates for discrete spaces with piecewise polynomials of degree . Building on techniques recently developed by Kovács, Li, Lubich, and collaborators for closed surfaces, we extend these ideas to surfaces with boundaries by formulating appropriate boundary conditions for both the mean curvature and the normal vector. These boundary treatments are essential for proving convergence. The core of our analysis involves a classical error splitting strategy using auxiliary discrete functions that approximate the surface geometry, the mean curvature, and the normal vector. We estimate two types of errors for each variable to rigorously assess both stability and consistency. To effectively handle boundary conditions for the normal vector, we introduce a nonlinear Ritz projection into the analysis. As a result, we derive optimal error estimates for the surface position, velocity, mean curvature, and normal vector. Our theoretical findings are corroborated by numerical experiments.

Paper Structure

This paper contains 10 sections, 2 theorems, 34 equations.

Table of Contents

  1. Introduction
  2. Main Challenges.
  3. Main Contributions.
  4. Notation for evolving surfaces
  5. Pullback, push-forward and transport functions.
  6. Discrete setting: B-spline based Isogeometric Analysis
  7. Vanishing boundary values.
  8. Mean curvature flow with Dirichlet boundary conditions
  9. Evolution equations for geometric quantities and boundary conditions
  10. Weak formulation

Key Result

lemma thmcounterlemma

Let $\mathcal{Q}: L^1 (\Omega)\rightarrow \mathbb{S}_{h}$ be the projector operator defined as in def: projector. Then, if $1\le q\le \infty$, $T>0$ and $U:\Omega\times[0,T]\to \mathbb{R}$, satisfies $U(t),\partial_t U(t)\in W^{l,q}(\Omega)$ for all $0\le t \le T$, we have that, for $0\leq k \leq l for a constant $C$, which may depend on $k$, $l$, $q$ and $p$, but is otherwise independent of $h$

Theorems & Definitions (5)

  • lemma thmcounterlemma
  • proof
  • lemma thmcounterlemma
  • proof
  • remark thmcounterremark