An energy-stable parametric finite element method for the planar Willmore flow

TL;DR

This work introduces an energy-stable parametric finite element method for planar Willmore flow by deriving two new geometric identities that couple the normal velocity with the normal and describe the curvature evolution. It reformulates the flow into a new geometric PDE with a mixed differential-algebraic structure and a separate curvature evolution equation, then provides a variational formulation that guarantees energy dissipation. The authors develop both semi-discrete and fully discrete PFEM schemes, prove unconditional energy stability, and demonstrate robust second-order convergence and mesh-quality preservation through extensive numerical tests. The approach yields a stable, efficient framework for simulating Willmore-type geometric evolutions, with potential extensions to related flows and improved mesh strategies.

Abstract

We propose an energy-stable parametric finite element method (PFEM) for the planar Willmore flow and establish its unconditional energy stability of the full discretization scheme. The key lies in the introduction of two novel geometric identities to describe the planar Willmore flow: the first one involves the coupling of the outward unit normal vector $\boldsymbol{n}$ and the normal velocity $V$, and the second one concerns the time derivative of the mean curvature $κ$. Based on them, we derive a set of new geometric partial differential equations for the planar Willmore flow, leading to our new fully-discretized and unconditionally energy-stable PFEM. Our stability analysis is also based on the two new geometric identities. Extensive numerical experiments are provided to illustrate its efficiency and validate its unconditional energy stability.

Figures (4)

• Figure 5.1: Plot of numerical errors $e^h$ of different initial curves at $t_m=1$: (a) a uniformly partitioned unit circle, (b) a non-uniformly partitioned unit circle, (c): an ellipse, (d): a 3-fold curve
• Figure 5.2: Plot of the number of iterations of different initial curves with $h=2^{-8}$ and $\tau=\frac{h^2}{2}$ at $t_m=1$: (a) a uniformly partitioned unit circle, (b) a non-uniformly partitioned unit circle, (c): an ellipse, (d): a 3-fold curve.
• Figure 5.3: Plot of the normalized energy $W^m/W^0$ with fixed mesh size $h = 2^{-8}$ and varying time step size $\tau = 0.0001, 0.001, 0.005$ for (a) the elliptical curves, and (b) the 3-fold curve.
• Figure 5.4: The evolution of (a) an initial elliptical curve (b) an initial 3-fold curve under the Willmore flow, where the initial curve is colored in red, and the intermediate curves evolve over time from dark blue to light blue. The mesh size $h = 2^{-5}$ and the time step size $\tau = 0.001$.

Theorems & Definitions (11)

• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Remark 2.1
• Theorem 2.3: Energy dissipation
• proof
• Theorem 3.1: Energy dissipation
• proof
• Theorem 4.1: Unconditional energy dissipation