Stable approximations for axisymmetric Willmore flow for closed and open surfaces^★

¹ Department of Mathematics, Imperial College, London SW7 2AZ, UK
² Fakultät für Mathematik, Universität Regensburg, 93040 Regensburg, Germany
³ Department of Mathematics, University of Trento, Trento, Italy

^* Corresponding author: robert.nurnberg@unitn.it

Received: 4 November 2019
Accepted: 12 March 2021

Abstract

For a hypersurface in ℝ³, Willmore flow is defined as the L²-gradient flow of the classical Willmore energy: the integral of the squared mean curvature. This geometric evolution law is of interest in differential geometry, image reconstruction and mathematical biology. In this paper, we propose novel numerical approximations for the Willmore flow of axisymmetric hypersurfaces. For the semidiscrete continuous-in-time variants we prove a stability result. We consider both closed surfaces, and surfaces with a boundary. In the latter case, we carefully derive weak formulations of suitable boundary conditions. Furthermore, we consider many generalizations of the classical Willmore energy, particularly those that play a role in the study of biomembranes. In the generalized models we include spontaneous curvature and area difference elasticity (ADE) effects, Gaussian curvature and line energy contributions. Several numerical experiments demonstrate the efficiency and robustness of our developed numerical methods.