A Stabilized Parametric Finite Element Method for Surface Diffusion with an Arbitrary Surface Energy

TL;DR

This work develops a structure-preserving stabilized parametric finite element method (SPFEM) for simulating anisotropic surface diffusion of curves with arbitrary surface energy $\hat{\gamma}(\theta)$. By introducing a nonnegative stabilizing function $k(\theta)$ and a stabilized surface energy matrix $\hat{\boldsymbol{G}}_k(\theta)$, the authors derive a conservative weak formulation and implement a SPFEM that preserves area and unconditionally dissipates energy. A rigorous framework is built around a minimal stabilizing function $k_0(\theta)$, including a local energy estimate and a proof of existence for $k_0(\theta)$, ensuring stability under mild anisotropy conditions like $3\hat{\gamma}(\theta)\ge\hat{\gamma}(\theta-\pi)$. The method extends to solid-state dewetting with contact-line migration and remains effective for globally $C^1$ and piecewise $C^2$ anisotropies, supported by extensive numerical results showing accuracy, stability, and structure preservation. This approach broadens the applicability of energy-stable PFEMs to general anisotropies and provides a practical tool for materials science applications such as thin-film dewetting and morphological evolution.

Abstract

We proposed a structure-preserving stabilized parametric finite element method (SPFEM) for the evolution of closed curves under anisotropic surface diffusion with an arbitrary surface energy $\hatγ(θ)$. By introducing a non-negative stabilizing function $k(θ)$ depending on $\hatγ(θ)$, we obtained a novel stabilized conservative weak formulation for the anisotropic surface diffusion. A SPFEM is presented for the discretization of this weak formulation. We construct a comprehensive framework to analyze and prove the unconditional energy stability of the SPFEM under a very mild condition on $\hatγ(θ)$. This method can be applied to simulate solid-state dewetting of thin films with arbitrary surface energies, which are characterized by anisotropic surface diffusion and contact line migration. Extensive numerical results are reported to demonstrate the efficiency, accuracy and structure-preserving properties of the proposed SPFEM with anisotropic surface energies $\hatγ(θ)$ arising from different applications.

Figures (13)

• Figure 1: An illustration of a closed curve under anisotropic surface diffusion with an anisotropic 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: An illustration of solid-state dewetting of a thin film on a flat rigid substrate in 2D, where $\gamma_{FV}=\hat{\gamma}(\theta),\gamma_{VS},\gamma_{FS}$ represent surface energy densities of film/vapor, vapor/substrate and film/substrate interface, respectively, $x_c^l,x_c^r$ are the left and right contact points.
• Figure 3: Convergence rates of the SPFEM \ref{['eqn:sp-pfem surface diffusion']} with $k(\theta)=k_0(\theta)$ for: (a) anisotropy in Case I at $t=0.5$ with different $\beta$; and (b) anisotropy in Case II with $b=-0.8$ at different times $t=0.125,0.25,0.5$.
• Figure 4: Weighted mesh ratio of the SPFEM \ref{['eqn:sp-pfem surface diffusion']} with $k(\theta)=k_0(\theta)$ for: (a) anisotropy in Case I with $\beta=\frac{1}{9}$; and (b) anisotropy in Case II with $b=-0.8$.
• Figure 5: Normalized area loss (blue dashed line) and iteration number (red line) of the SPFEM \ref{['eqn:sp-pfem surface diffusion']} with $k(\theta)=k_0(\theta)$ and $h=2^{-7},\tau=2^{-10}$ for: (a) anisotropy in Case I with $\beta=\frac{1}{2}$; and (b) anisotropy in Case II with $b=-0.8$.

Theorems & Definitions (31)

• Theorem 2.1
• proof
• Remark 2.1
• Remark 2.2
• Proposition 3.1: area conservation and energy dissipation
• Lemma 3.1
• proof
• proof
• Proposition 3.2: area conservation and energy dissipation
• Remark 3.1