Efficient Energy-Stable Parametric Finite Element Methods for Surface Diffusion Flow and Applications in Solid-State Dewetting (Communications in Computational Physics, 2025)
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 theoreticallyproven to be energy-stable poses a significant challenge. Motivated by the new scalarauxiliary variable approach [1], we propose novel energy-stable parametric finite element approximations for isotropic/anisotropic surface diffusion flows, achieving bothfirst-order and second-order accuracy in time. Additionally, we apply the algorithmsto 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 theycan be readily extended to other high-order time discretization methods (e.g., BDFkschemes). Meanwhile, the algorithms achieve remarkable computational efficiencyand maintain excellent mesh quality. More importantly, the algorithm can be theoretically proven to possess unconditional energy stability, with the energy nearly equal tothe original energy.