A unified structure-preserving parametric finite element method for anisotropic surface diffusion

Abstract

We propose and analyze a unified structure-preserving parametric finite element method (SP-PFEM) for the anisotropic surface diffusion of curves in two dimensions $(d=2)$ and surfaces in three dimensions $(d=3)$ with an arbitrary anisotropic surface energy density $γ(\boldsymbol{n})$, where $\boldsymbol{n}\in \mathbb{S}^{d-1}$ represents the outward unit vector. By introducing a novel unified surface energy matrix $\boldsymbol{G}_k(\boldsymbol{n})$ depending on $γ(\boldsymbol{n})$, the Cahn--Hoffman $\boldsymbolξ$-vector and a stabilizing function $k(\boldsymbol{n}):\ \mathbb{S}^{d-1}\to {\mathbb R}$, we obtain a unified and conservative variational formulation for the anisotropic surface diffusion via different surface differential operators including the surface gradient operator, the surface divergence operator and the surface Laplace--Beltrami operator. A SP-PFEM discretization is presented for the variational problem. In order to establish the unconditional energy stability of the proposed SP-PFEM under a very mild condition on $γ(\boldsymbol{n})$, we propose a new framework via {\sl local energy estimate} for proving energy stability/structure-preserving properties of the parametric finite element method for the anisotropic surface diffusion. This framework sheds light on how to prove unconditional energy stability of other numerical methods for geometric partial differential equations. Extensive numerical results are reported to demonstrate the efficiency and accuracy as well as structure-preserving properties of the proposed SP-PFEM for the anisotropic surface diffusion with arbitrary anisotropic surface energy density $γ(\boldsymbol{n})$ arising from different applications.

Figures (8)

• Figure 3.1: Plot of the direction vector $\mathcal{J}$ for 2D (left) and for 3D (right).
• Figure 4.1: Illustration of the local energy estimate Lemma \ref{['lem: local est']} for 2D (left) and for 3D (right).
• Figure 6.1: Plot of the normalized volume change $\frac{\Delta V(t)}{V(0)}$ for different cases: (a) for Case I, (b) for Case II, and (c) for Case III.
• Figure 6.2: Plot of the normalized energy $\frac{W(t)}{W(0)}$ for anisotropic energies in Case I-III with the fixed $k(\boldsymbol{n})=k_0(\boldsymbol{n})$ (left column) for different $h$ and $\tau$; or the fixed $h=2^{-4}$ and $\tau=\frac{2}{25}h^2$ with different $k(\boldsymbol{n})$ (right column). The top, middle, and bottom rows correspond to the anisotropic energies in Case I-III, respectively.
• Figure 6.3: Plot of the number of iterations per time step for anisotropies in (a) Case I, (b) Case II, and (c) Case III.

Theorems & Definitions (33)

• Definition 2.1: Surface energy matrix
• Theorem 2.2: Strong formulation of $\mu$
• proof
• Theorem 2.3: Weak formulation of $\mu$
• proof
• Proposition 2.4
• Remark 3.1
• Theorem 3.2: structure-preserving
• Lemma 4.1: local energy estimate
• Remark 4.2