Stable Backward Differentiation Formula Time Discretization of BGN-Based Parametric Finite Element Methods for Geometric Flows

SIAM Journal on Scientific Computing, Volume 46, Issue 5, Page A2874-A2898, October 2024.
Abstract. We propose a novel class of temporal high-order parametric finite element methods for solving a wide range of geometric flows of curves and surfaces. By incorporating the backward differentiation formula (BDF) for time discretization into the BGN formulation, originally proposed by Barrett, Garcke, and Nürnberg (J. Comput. Phys., 222 (2007), pp. 441–467), we successfully develop high-order BGN/BDF[math] schemes. The proposed BGN/BDF[math] schemes not only retain almost all the advantages of the classical first-order BGN scheme such as computational efficiency and good mesh quality, but also exhibit the desired [math]th-order temporal accuracy in terms of shape metrics, ranging from second-order to fourth-order accuracy. Furthermore, we validate the performance of our proposed BGN/BDF[math] schemes through extensive numerical examples, demonstrating their high-order temporal accuracy for various types of geometric flows while maintaining good mesh quality throughout the evolution.