Efficient energy-stable parametric finite element methods for surface diffusion flow and applications in solid-state dewetting

TL;DR

This paper introduces high-order, energy-stable parametric finite element methods for surface diffusion flows, validated through simulations of solid-state dewetting, demonstrating improved accuracy, stability, and efficiency over existing methods.

Contribution

The paper develops novel high-order energy-stable finite element algorithms for geometric flows, extending the scalar auxiliary variable approach to achieve second-order accuracy.

Findings

Algorithms are energy-stable and highly accurate.

Numerical experiments confirm efficiency and mesh quality.

Methods are adaptable to other high-order schemes.

Abstract

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.