Stable BDF time discretization of BGN-based parametric finite element methods for geometric flows
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 formulae (BDF) for time discretization into the BGN formulation, originally proposed by Barrett, Garcke, and N\"urnberg (J. Comput. Phys., 222 (2007), pp.~441--467), we successfully develop high-order BGN/BDF schemes. The proposed BGN/BDF 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 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 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.
Cite
@article{arxiv.2402.03641,
title = {Stable BDF time discretization of BGN-based parametric finite element methods for geometric flows},
author = {Wei Jiang and Chunmei Su and Ganghui Zhang},
journal= {arXiv preprint arXiv:2402.03641},
year = {2024}
}