论文标题
对对称的参数有限元法，用于3D中各向异性表面扩散
A symmetrized parametric finite element method for anisotropic surface diffusion in 3D
论文作者
论文摘要
对于在各向异性表面扩散下的闭合表面在三维（\ boldsymbol {n}）$中的一般表面扩散（3D），其中$ \ boldsymbol {n} $是单位外向载体，是通过引入一个新型的对称的正常载体 $\boldsymbol{Z}_k(\boldsymbol{n})$ depending on a stabilizing function $k(\boldsymbol{n})$ and the Cahn-Hoffman $\boldsymbolξ$-vector, we present a new symmetrized variational formulation for anisotropic surface diffusion with weakly or strongly anisotropic surface energy, which preserves two重要结构，包括节约体积和能量耗散。然后，我们提出了一种结构性保护参数有限元法（SP-PFEM），以离散对称的变分问题，该问题将体积保留在离散的水平中。在$γ（\ boldsymbol {n}）$上的相对温和和简单的条件下，我们表明，对于几乎所有各向异性表面能$γ（\ boldsymbol {n}）$，SP-PFEM无条件能稳定。据报道，广泛的数值结果证明了拟议的SP-PFEM在3D中求解各向异性表面扩散的效率和准确性以及能量耗散。
For the evolution of a closed surface under anisotropic surface diffusion with a general anisotropic surface energy $γ(\boldsymbol{n})$ in three dimensions (3D), where $\boldsymbol{n}$ is the unit outward normal vector, by introducing a novel symmetric positive definite surface energy matrix $\boldsymbol{Z}_k(\boldsymbol{n})$ depending on a stabilizing function $k(\boldsymbol{n})$ and the Cahn-Hoffman $\boldsymbolξ$-vector, we present a new symmetrized variational formulation for anisotropic surface diffusion with weakly or strongly anisotropic surface energy, which preserves two important structures including volume conservation and energy dissipation. Then we propose a structural-preserving parametric finite element method (SP-PFEM) to discretize the symmetrized variational problem, which preserves the volume in the discretized level. Under a relatively mild and simple condition on $γ(\boldsymbol{n})$, we show that SP-PFEM is unconditionally energy-stable for almost all anisotropic surface energies $γ(\boldsymbol{n})$ arising in practical applications. Extensive numerical results are reported to demonstrate the efficiency and accuracy as well as energy dissipation of the proposed SP-PFEM for solving anisotropic surface diffusion in 3D.
