An energy-stable parametric finite element method for Willmore flow with normal-tangential velocity splitting

TL;DR

The paper introduces a first-of-its-kind energy-stable, fully discrete parametric finite element method for Willmore flow of hypersurfaces in 2D and 3D. By splitting normal and tangential velocities via a novel geometric PDE that combines a curvature evolution equation with a curvature identity, the authors preserve the gradient-flow structure and achieve unconditional discrete energy decay. The method is linear, decouples into two small linear systems per time step, and accommodates spontaneous curvature and open surfaces with boundary through a flexible weak formulation and boundary-condition framework. Theoretical results establish well-posedness and unconditional energy stability, while extensive 2D and 3D numerical experiments demonstrate accuracy, robustness, and mesh-quality preservation, validating the approach for both closed and open geometries with various boundary conditions.

Abstract

We propose and analyze an energy-stable fully discrete parametric approximation for Willmore flow of hypersurfaces in two and three space dimensions. We allow for the presence of spontaneous curvature effects and for open surfaces with boundary. The presented scheme is based on a new geometric partial differential equation (PDE) that combines an evolution equation for the mean curvature with a separate equation that prescribes the tangential velocity. The mean curvature is used to determine the normal velocity within the gradient flow structure, thus guaranteeing an unconditional energy stability for the discrete solution upon suitable discretization. We introduce a novel weak formulation for this geometric PDE, in which different types of boundary conditions can be naturally enforced. We further discretize the weak formulation to obtain a fully discrete parametric finite element method, for which well-posedness can be rigorously shown. Moreover, the constructed scheme admits an unconditional stability estimate in terms of the discrete energy. Extensive numerical experiments are reported to showcase the accuracy and robustness of the proposed method for computing Willmore flow of both curves in $\mathbb{R}^2$ and surfaces in $\mathbb{R}^3$.

Figures (15)

• Figure 2.1: Sketch of ${\Gamma(t)}$ with boundary $\partial{\Gamma(t)}$, as well as the three unit vectors $\vec{\tau},\vec{\mu}$ and $\vec{\nu}$ which form an orthonormal basis of ${\mathbb R}^3$.
• Figure 5.1: Numerical errors for an expanding/shrinking circle with ${\overline{\varkappa}}=-0.5$ (left panel) and ${\overline{\varkappa}}=-2$ (right panel), where $\Delta t=(\frac{2^{5}\,h}{5})^2$.
• Figure 5.2: [${\overline{\varkappa}}=-2$] Numerical errors in the evolution of an initial circle segment under Navier boundary conditions (left panel) and clamped boundary conditions (right panel), where $\Delta t=(\frac{2^{5}\,h}{5})^2$.
• Figure 5.3: [${\overline{\varkappa}}=-0.5$] Evolution of an initial circle segment towards the steady state (red line) under different boundary conditions. We plot $\Gamma^m$ at times $t=0,2,4, \cdots, 20, 50$. On the bottom are plots of the discrete energy and the mesh ratio ${\rm R}^m$.
• Figure 5.4: [${\overline{\varkappa}}=-2$] Evolution of an initial circle segment towards the steady state (red line) under different boundary conditions. We plot $\Gamma^m$ at times $t=0,0.1,\cdots,0.6, 2$. On the bottom are plots of the discrete energy and the mesh ratio ${\rm R}^m$.

Theorems & Definitions (13)

• Lemma 2.1
• Remark 2.2
• Lemma 3.1
• proof
• Theorem 3.2
• proof
• Lemma 4.1
• proof
• Remark 4.1
• Theorem 4.2: existence and uniqueness