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.

翻译：本文提出了一种新颖的参数有限元方法，用于模拟曲线和曲面的表面扩散过程。我们的核心策略采用了预测-校正时间步进法，将经典的一阶时间精度提升至二阶精度。值得注意的是，新方法无需网格正则化技术，这使其有别于作者先前提出的二阶格式（J. Comput. Phys. 514 (2024) 113220）。此外，该方法保持了一阶格式的长期网格均匀分布特性。所提出的技术可轻松推广至其他几何流问题，例如（保面积的）曲线缩短流以及具有各向异性表面能的表面扩散。我们通过系统的数值实验验证了所提方法的精度与效率，结果证明其性能优于现有方案。