Stable fully practical finite element methods for axisymmetric Willmore flow

TL;DR

This paper introduces unconditionally stable, fully discrete finite element schemes for axisymmetric Willmore flow that do not require remeshing, using a novel weak formulation involving the generating curve's curvature.

Contribution

The paper proposes a new weak formulation and two fully discrete schemes for axisymmetric Willmore flow that are unconditionally stable and do not require remeshing.

Findings

Schemes are unconditionally stable and reliable without remeshing.

Numerical results confirm convergence, stability, and equidistribution.

The approach effectively handles spontaneous curvature effects.

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.