Predictor-corrector, BGN-based parametric finite element methods for surface diffusion

TL;DR

This paper introduces a predictor-corrector finite element method for surface diffusion that achieves second-order accuracy without mesh regularization, maintains mesh quality, and is adaptable to various geometric flows.

Contribution

The authors develop a novel predictor-corrector scheme for parametric finite element methods that improves accuracy and stability in simulating surface diffusion without mesh regularization.

Findings

Achieves second-order temporal accuracy in surface diffusion simulations.

Eliminates the need for mesh regularization techniques.

Demonstrates superior accuracy and efficiency in numerical experiments.

Abstract

We present a novel parametric finite element approach for simulating the surface diffusion of curves and surfaces. Our core strategy incorporates a predictor-corrector time-stepping method, which enhances the classical first-order temporal accuracy to achieve second-order accuracy. Notably, our new method eliminates the necessity for mesh regularization techniques, setting it apart from previously proposed second-order schemes by the authors (J. Comput. Phys. 514 (2024) 113220). Moreover, it maintains the long-term mesh equidistribution property of the first-order scheme. The proposed techniques are readily adaptable to other geometric flows, such as (area-preserving) curve shortening flow and surface diffusion with anisotropic surface energy. Comprehensive numerical experiments have been conducted to validate the accuracy and efficiency of our proposed methods, demonstrating their superiority over previous schemes.