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For the evolution of a closed surface under anisotropic surface diffusionwith a general anisotropic surface energy $\gamma(\boldsymbol{n})$ in threedimensions (3D), where $\boldsymbol{n}$ is the unit outward normal vector, byintroducing 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 $\boldsymbol{\xi}$-vector, we presenta new symmetrized variational formulation for anisotropic surface diffusionwith weakly or strongly anisotropic surface energy, which preserves twoimportant structures including volume conservation and energy dissipation. Thenwe propose a structural-preserving parametric finite element method (SP-PFEM)to discretize the symmetrized variational problem, which preserves the volumein the discretized level. Under a relatively mild and simple condition on$\gamma(\boldsymbol{n})$, we show that SP-PFEM is unconditionally energy-stablefor almost all anisotropic surface energies $\gamma(\boldsymbol{n})$ arising inpractical applications. Extensive numerical results are reported to demonstratethe efficiency and accuracy as well as energy dissipation of the proposedSP-PFEM for solving anisotropic surface diffusion in 3D.

A symmetrized parametric finite element method for anisotropic surface diffusion in 3D