An Energy-Stable Parametric Finite Element Method for Willmore Flow with Normal-Tangential Velocity Splitting

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 $\mathbb{R}^3$.