A structure-preserving parametric finite element method with optimal energy stability condition for anisotropic surface diffusion

TL;DR

The paper tackles the numerical simulation of anisotropic surface diffusion while preserving key geometric properties. It introduces a structure-preserving PFEM that uses a symmetric surface energy matrix and a stabilizing function to obtain a conservative weak formulation. The main theoretical contribution is proving that the optimal energy stability condition $3\hat{\gamma}(\theta)-\hat{\gamma}(\theta-\pi)\geq 0$ is necessary and sufficient for unconditional stability, along with existence and a global bound for the minimal stabilizing function $k_0(\theta)$. Numerical results demonstrate accurate area conservation, energy dissipation, and robust performance across symmetric and asymmetric anisotropies, including extensions to open curves in dewetting scenarios.

Abstract

We propose and analyze a structure-preserving parametric finite element method (SP-PFEM) for the evolution of closed curves under anisotropic surface diffusion with surface energy density $\hatγ(θ)$. Our primary theoretical contribution establishes that the condition $3\hatγ(θ)-\hatγ(θ-π)\geq 0$ is both necessary and sufficient for unconditional energy stability within the framework of local energy estimates. The proposed method introduces a symmetric surface energy matrix $\hat{\boldsymbol{Z}}_k(θ)$ with a stabilizing function $k(θ)$, leading to a conservative weak formulation. Its fully discretization via SP-PFEM rigorously preserves the two geometric structures: enclosed area conservation and energy dissipation unconditionally under our energy stability condition. Numerical results are reported to demonstrate the efficiency and accuracy of the proposed method, along with its area conservation and energy dissipation properties.

Figures (10)

• Figure 1: An illustration of a closed curve under anisotropic surface diffusion with surface energy $\hat{\gamma}(\theta)$, while $\theta$ is the angle between the $y$-axis and the unit outward normal vector $\boldsymbol{n}=\boldsymbol{n}(\theta)\coloneqq(-\sin\theta,\cos\theta)^T$. $\boldsymbol{\tau}=\boldsymbol{\tau}(\theta)\coloneqq(\cos\theta,\sin\theta)^T$ represents the unit tangent vector.
• Figure 2: Minimal stabilizing function $k_0(\theta)$ and upper bound $K(\theta)$ in \ref{['eqn:upper bound']} for Case I with $\beta=\frac{1}{2}$.
• Figure 3: Convergence rates of the SP-PFEM \ref{['eqn:SP-PFEM']} for Case I with $\beta=1/9$ (a) at different times with $k(\theta)=k_0(\theta)$, and (b) at $t=0.5$ with different $k(\theta)$; and for Case II (c) at different times with $k(\theta)=k_0(\theta)$, and (d) at $t=0.5$ with different $k(\theta)$.
• Figure 4: Normalized area loss (blue dash line) and iteration number (red line) of the SP-PFEM \ref{['eqn:SP-PFEM']} with $k(\theta)=k_0(\theta)$ and $h=2^{-6},\tau=2^{-12}$ for (a) Case I with $\beta=1/2$; and for (b) Case II.
• Figure 5: Normalized energy of the SP-PFEM \ref{['eqn:SP-PFEM']} with $k(\theta)=k_0(\theta)$ for (a) Case I with $\beta=1/2$; and for (b) Case II.

Theorems & Definitions (22)

• definition thmcounterdefinition: symmetric surface energy matrix
• theorem 2.1
• proof
• remark thmcounterremark
• proposition thmcounterproposition: area conservation and energy dissipation
• remark thmcounterremark
• theorem 3.1: area conservation and unconditional energy stability
• remark thmcounterremark
• theorem 4.1
• remark thmcounterremark