An energy-stable parametric finite element method for the Willmore flow in three dimensions

TL;DR

This paper introduces a novel energy-stable parametric finite element method for simulating the Willmore flow in three dimensions, ensuring energy dissipation and improved mesh quality during surface evolution.

Contribution

The work develops the first energy-stable PFEM for the Willmore flow, incorporating new geometric identities and tangential velocity control for enhanced robustness and mesh quality.

Findings

Method preserves energy dissipation during simulations.

Achieves high mesh quality in surface evolution.

Successfully applied to Gauss curvature flow.

Abstract

This work develops novel energy-stable parametric finite element methods (ES-PFEM) for the Willmore flow and curvature-dependent geometric gradient flows of surfaces in three dimensions. The key to achieving the energy stability lies in the use of two novel geometric identities: (i) a reformulated variational form of the normal velocity field, and (ii) incorporation of the temporal evolution of the mean curvature into the governing equations. These identities enable the derivation of a new variational formulation. By using the parametric finite element method, an implicit fully discrete scheme is subsequently developed, which maintains the energy dissipative property at the fully discrete level. Based on the ES-PFEM, comprehensive insights into the design of ES-PFEM for general curvature-dependent geometric gradient flows and a new understanding of mesh quality improvement in PFEM are provided. In particular, we develop the first PFEM for the Gauss curvature flow of surfaces. Furthermore, a tangential velocity control methodology is applied to improve the mesh quality and enhance the robustness of the proposed numerical method. Extensive numerical experiments confirm that the proposed method preserves energy dissipation properties and maintain good mesh quality in the surface evolution under the Willmore flow.