Stable fully practical finite element methods for axisymmetric Willmore flow

TL;DR

The paper tackles axisymmetric Willmore flow with spontaneous curvature by developing fully discrete finite element schemes that preserve the gradient-flow energy dissipation and maintain mesh quality without remeshing. It introduces a novel weak formulation that couples an evolution equation for the surface mean curvature with a curvature-based representation of the generating curve, ensuring an unconditional energy stability. Two fully discrete schemes (linear and nonlinear) are proven energy-stable and incorporate a tangential (BGN-type) motion to achieve equidistribution of mesh points, with a semidiscrete counterpart that also guarantees dissipation and equidistribution. Numerical experiments on genus-0 and genus-1 surfaces validate convergence orders, energy decay, and robust mesh behavior, including convergence toward the Clifford torus and observed pinch-off phenomena under certain spontaneous curvature settings.

Abstract

We consider fully discrete numerical approximations for axisymmetric Willmore flow that are unconditionally stable and work reliably without remeshing. We restrict our attention to surfaces without boundary, but allow for spontaneous curvature effects. The axisymmetric setting allows us to formulate our schemes in terms of the generating curve of the considered surface. We propose a novel weak formulation, that combines an evolution equation for the surface's mean curvature and the curvature identity of the generating curve. The mean curvature is used to describe the gradient flow structure, which enables an unconditional stability result for the discrete solutions. The generating curve's curvature, on the other hand, describes the surface's in-plane principal curvature and plays the role of a Lagrange multiplier for an equidistribution property on the discrete level. We introduce two fully discrete schemes and prove their unconditional stability. Numerical results are provided to confirm the convergence, stability and equidistribution properties of the introduced schemes.

Figures (10)

• Figure 1: Sketch of $\Gamma$ and $\mathcal{S}$, as well as the unit vectors $\vec{e}_1$, $\vec{e}_2$ and $\vec{e}_3$.
• Figure 2: [${\overline{\varkappa}}=0$, $J = 128$, $\Delta t = 10^{-3}$] Evolution of an initial disk of dimension of $7\times 1\times 7$. We plot $\Gamma^m$ at times $t=0,0.5,1,\cdots,3, 10$ and visualize the axisymmetric surfaces at $t=0.5$ and $t=10$.
• Figure 3: Plots of the discrete energy and the mesh ratio ${\rm R}^m$ for the evolution of the disk in Fig. \ref{['fig:disk1']}.
• Figure 4: [${\overline{\varkappa}}=-1.25$, $J = 128$, $\Delta t = 10^{-3}$] Evolution of an initial disk of dimension of $7\times 1\times 7$. We plot $\Gamma^m$ at times $t=0,0.5,1,\cdots, 3,10$ and visualize the axisymmetric surfaces at $t=0.5$ and $t=10$.
• Figure 5: Plots of the discrete energy and the mesh ratio ${\rm R}^m$ for the evolution of the disk in Fig. \ref{['fig:disk2']}.

Theorems & Definitions (20)

• Lemma 2.1
• proof
• Remark 2.2
• Lemma 2.3
• proof
• Remark 2.4
• Theorem 2.5
• proof
• Theorem 3.1
• proof