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 $\gamma(\boldsymbol{n})$, 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 $\gamma(\boldsymbol{n})$, the Cahn--Hoffman $\boldsymbol{\xi}$-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 $\gamma(\boldsymbol{n})$, 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 $\gamma(\boldsymbol{n})$ arising from different applications.

翻译：针对二维$(d=2)$曲线与三维$(d=3)$曲面在各向异性表面能密度$\gamma(\boldsymbol{n})$（其中$\boldsymbol{n}\in \mathbb{S}^{d-1}$表示外法向单位向量）作用下的各向异性表面扩散问题，我们提出并分析了一种统一的结构保持参数有限元方法（SP-PFEM）。通过引入依赖于$\gamma(\boldsymbol{n})$的新型统一表面能矩阵$\boldsymbol{G}_k(\boldsymbol{n})$、Cahn--Hoffman $\boldsymbol{\xi}$向量及稳定化函数$k(\boldsymbol{n}):\ \mathbb{S}^{d-1}\to {\mathbb R}$，并利用包括表面梯度算子、表面散度算子和表面Laplace--Beltrami算子在内的不同表面微分算子，我们建立了各向异性表面扩散的统一守恒变分形式。针对该变分问题提出了SP-PFEM离散格式。为在$\gamma(\boldsymbol{n})$满足极弱条件下证明所提SP-PFEM的无条件能量稳定性，我们建立了基于{\sl局部能量估计}的新框架，用于验证参数有限元方法在各向异性表面扩散中的能量稳定/结构保持特性。该框架为几何偏微分方程其他数值方法无条件能量稳定性的证明提供了理论支撑。大量数值实验表明，所提出的SP-PFEM对于源自不同应用场景且具有任意各向异性表面能密度$\gamma(\boldsymbol{n})$的各向异性表面扩散问题，具备高效性、精确性及结构保持特性。