[Paper Review] Finite element approximation of power mean curvature flow

Heiko Kröner|arXiv (Cornell University)|Aug 11, 2013

Geometric Analysis and Curvature Flows9 references3 citations

TL;DR

This paper presents a finite element method for approximating power mean curvature flow (PMCF) in $ ^{n+1}$, using a regularized level set formulation with $C^0$ finite elements of degree $\ u \leq 2$. For $n=1$, it establishes a convergence rate of the form $\|u - u_h^\epsilon\|_{C^{0,\Theta}} \leq c\epsilon^\lambda + c\epsilon^{-\gamma}h^\delta$, proving polynomial convergence in both regularization parameter $\epsilon$ and mesh size $h$.

ABSTRACT

In [21] the evolution of hypersurfaces in $\mathbb{R}^{n+1}$ with normal speed equal to a power $k>1$ of the mean curvature is considered and the levelset solution $u$ of the flow is obtained as the $C^0$-limit of a sequence $u^ε$ of smooth functions solving the regularized levelset equations. We prove a rate for this convergence. Then we triangulate the domain by using a tetraeder mesh and consider continuous finite elements, which are polynomials of degree $\le 2$ on each tetraeder of the triangulation. We show in the case $n=1$ (i.e. the evolving hypersurfaces are curves), that there are solutions $u^ε_h$ of the above regularized equations in the finite element sense, and estimate the approximation error between $u^ε_h$ and $u$. Our method can be extended to the case $n>1$, if one uses higher order finite elements.

Motivation & Objective

• To develop a numerical scheme for approximating power mean curvature flow (PMCF), a nonlinear geometric evolution where normal speed is a power $k>1$ of mean curvature.
• To establish convergence rates for the finite element approximation of the level set solution of PMCF, using a regularized level set formulation as an intermediate step.
• To analyze the error between the exact viscosity solution $u$, the regularized solution $u^\epsilon$, and the finite element solution $u_h^\epsilon$, particularly in Hölder norms.
• To extend the method to higher dimensions ($n>1$) using higher-order finite elements, with theoretical justification for convergence rates.

Proposed method

• Formulate PMCF via a level set equation: $\text{div}\left(\frac{Du}{|Du|}\right) = -|Du|^{-1/k}$ in $\Omega$, with $u=0$ on $\partial\Omega$, where $u$ is the viscosity solution.
• Introduce a regularized level set equation: $\text{div}\left(\frac{Du^\epsilon}{\sqrt{\epsilon^2 + |Du^\epsilon|^2}}\right) = - (\epsilon^2 + |Du^\epsilon|^2)^{-1/(2k)}$, ensuring smooth solutions $u^\epsilon$ for small $\epsilon>0$.
• Discretize the domain using a tetrahedral mesh $\mathbb{T}_h$, and define continuous $C^0$ finite elements of degree $\leq 2$ on each tetrahedron.
• Construct a finite element solution $u_h^\epsilon$ by solving the regularized equation in the finite element space, using Galerkin or weak formulation.
• Apply error decomposition: $\|u - u_h^\epsilon\|_{C^{0,\Theta}} \leq \|u - u^\epsilon\|_{C^{0,\Theta}} + \|u^\epsilon - u_h^\epsilon\|_{C^{0,\Theta}}$, with the first term estimated via viscosity solution theory and the second via finite element approximation theory.
• Use inverse estimates, interpolation error bounds, and stability estimates in $L^\mu$ and $H^{1,\mu}$ norms to control the finite element error, with careful tracking of constants depending on $\epsilon$ and $h$.

Experimental results

Research questions

• RQ1 What is the convergence rate of the finite element approximation to the viscosity solution of power mean curvature flow in the $C^{0,\Theta}$ norm for $n=1$?
• RQ2 How do the regularization parameter $\epsilon$ and mesh size $h$ interact to yield a polynomial convergence rate in the finite element approximation of PMCF?
• RQ3 Can the error between the exact solution $u$, the regularized solution $u^\epsilon$, and the finite element solution $u_h^\epsilon$ be bounded with explicit dependence on $\epsilon$ and $h$?
• RQ4 Is it possible to extend the finite element method to higher dimensions ($n>1$) with higher-order elements while maintaining a polynomial convergence rate?
• RQ5 What role does the level set formulation with fixed time dependence play in simplifying the nonlinearity from the exponent $k>1$ in the curvature term?

Key findings

• For $n=1$, the paper establishes a convergence rate of the form $\|u - u_h^\epsilon\|_{C^{0,\Theta}} \leq c\epsilon^\lambda + c\epsilon^{-\gamma}h^\delta$ for any $0<\Theta<\frac{1}{2}$, with explicit positive constants $\lambda, \gamma, \delta$ depending on $\Theta$ and $k$.
• The error between the viscosity solution $u$ and the regularized solution $u^\epsilon$ decays as $\|u - u^\epsilon\|_{C^{0,\Theta}} \leq c\epsilon^\lambda$, with $\lambda = \lambda(\Theta,k)$, proving the regularized solution converges to the true solution in Hölder norm.
• The finite element error $\|u^\epsilon - u_h^\epsilon\|_{C^{0,\Theta}}$ is bounded by $c\epsilon^{-\gamma}h^\delta$, showing that the mesh size $h$ must be chosen carefully relative to $\epsilon$ to maintain convergence.
• The analysis relies on $L^\mu$ and $H^{1,\mu}$ estimates for the finite element error, with inverse estimates and interpolation error bounds used to control the dependence on $h$ and $\epsilon$.
• For $n>1$, the method extends to higher-order finite elements, though the convergence rate analysis is carried out in detail only for $n=1$.
• The level set formulation with fixed time dependence ensures that the nonlinearity from the exponent $k>1$ affects only lower-order spatial derivatives, simplifying the analysis compared to time-dependent level set formulations.