An energy-stable parametric finite element approximation for axisymmetric Willmore flow of closed surfaces

Abstract

In this paper, we propose and analyze an energy-stable approximation for axisymmetric Willmore flow of closed surfaces. This approach extends the original work of Bao and Li [4] for the planar Willmore flow of curves. Through relations among various geometric quantities, we derive a system of equivalent geometric equations for the axisymmetric Willmore flow, including the evolution equations for the parameterization and mean curvature. The proposed method consists of the linear parametric finite element method in space and the backward Euler method in time. Furthermore, we prove that the fully discrete scheme is unconditionally energy-stable. The Newton-Raphson iteration method is adopted to solve the nonlinear system. Finally, numerical examples are presented to illustrate the efficiency and energy stability of the proposed method for Willmore flow in an axisymmetric setting.