用于表面扩散流的高效能量稳定参数化有限元方法及其在固态去湿中的应用
Efficient energy-stable parametric finite element methods for surface diffusion flow and applications in solid-state dewetting
AI总结：
针对现有表面扩散流能量稳定参数化有限元方法时间精度低的问题，基于标量辅助变量方法提出兼具一、二阶时间精度的能量稳定格式，可扩展至其他高阶离散，经实验验证精度、稳定性与效率，适用于固态去湿模拟。
摘要：
目前已有的用于表面扩散流及其他流动的能量稳定参数化有限元方法通常在时间上仅局限于一阶精度。设计一种同时具备理论可证能量稳定性的几何流动高阶算法是一项重大挑战。受新型标量辅助变量方法[F.Huang, J.Shen, Z.Yang, SIAM J. SCI. Comput., 42 (2020), pp. A2514-A2536]的启发，我们针对各向同性/各向异性表面扩散流提出了新型能量稳定参数化有限元近似格式，在时间上可实现一阶和二阶精度。此外，我们将这些算法应用于模拟薄膜的固态去湿过程。最后，大量数值实验验证了我们所开发数值方法的精度、能量稳定性与效率。本工作设计的算法展现出很强的通用性，可轻松扩展至其他高阶时间离散方法（如BDFk格式）。同时，这些算法实现了显著的计算效率并保持了优异的网格质量。更重要的是，该算法理论上可证明具有无条件能量稳定性，其能量近似等于原始能量。
英文摘要：
Currently existing energy-stable parametric finite element methods for surface diffusion flow and other flows are usually limited to first-order accuracy in time. Designing a high-order algorithm for geometric flows that can also be theoretically proven to be energy-stable poses a significant challenge. Motivated by the new scalar auxiliary variable approach [F.Huang, J.Shen, Z.Yang, SIAM J. SCI. Comput., 42 (2020), pp. A2514-A2536], we propose novel energy-stable parametric finite element approximations for isotropic/anisotropic surface diffusion flows, achieving both first-order and second-order accuracy in time. Additionally, we apply the algorithms to simulate the solid-state dewetting of thin films. Finally, extensive numerical experiments validate the accuracy, energy stability, and efficiency of our developed numerical methods. The designed algorithms in this work exhibit strong versatility, as they can be readily extended to other high-order time discretization methods (e.g., BDFk schemes). Meanwhile, the algorithms achieve remarkable computational efficiency and maintain excellent mesh quality. More importantly, the algorithm can be theoretically proven to possess unconditional energy stability, with the energy nearly equal to the original energy.
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