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

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.

Figures (13)

• Figure 1.1: An illustration of SDF on a closed curve $\Gamma(t)$ with anisotropic surface energy density in two dimensions.
• Figure 5.1: Convergence rates of BDF1-SAV and BDF2-SAV at times $T = 0.1$, $1.5$ with different surface energy densities: $\gamma(\theta)\equiv 1$ and $\gamma(\theta)=1+0.05\cos(4\theta)$.
• Figure 5.4: Temporal evolution of the mesh ratio $\Psi(t)$ for the three schemes with different surface energies $\gamma(\theta)=1+\beta \cos(4\theta)$: $\beta = 0, \frac{1}{20}.$ Parameters are chosen as $N=2^{-7}$, $\Delta t = 10^{-3}$, $r = 3$.
• Figure 5.5: The area loss $\left|A^h(\Vec{X}^m)-A^h(\Vec{X}^0) \right|$ for the BDF1-CSAV at r = 2,3,4 under surface energy $\gamma(\theta) = 1+ \beta \cos(4\theta)$: $\beta =0,\frac{1}{20},\frac{1}{10}$. Parameters are chosen as $N = 80$, $\Delta t = \frac{1}{160}$.
• Figure 5.6: Temporal evolution of $\Delta W^h(t)$ for the three schemes with $\gamma(\theta)=1+\frac{1}{10}\cos(4\theta)$. Parameters are chosen as $N=640$, $r=3$.

Theorems & Definitions (11)

• Remark 1
• Theorem 2.1
• proof
• Remark 2
• Theorem 3.1
• proof
• Remark 3
• Theorem 3.2
• proof
• Remark 4