A Symmetrized Parametric Finite Element Method for Anisotropic Surface Diffusion in Three Dimensions

Abstract

.We extend the symmetrized parametric finite method for the evolution of a closed curve under anisotropic surface diffusion in two dimensions, recently proposed by us [W. Bao, W. Jiang, and Y. Li, SIAM J. Numer. Anal., 61 (2023), pp. 617–641], to the evolution of a closed and orientable surface under anisotropic surface diffusion with a general anisotropic surface energy \(\gamma (\boldsymbol{n})\) in three dimensions (3D), where \(\boldsymbol{n}\in{\mathbb S}^2\) 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 \(k(\boldsymbol{n}):\ {\mathbb S}^2\to{\mathbb R}\) and the Cahn–Hoffman \(\boldsymbol{\xi }\) -vector, we present a new symmetrized variational formulation for anisotropic surface diffusion in 3D with weakly or strongly anisotropic surface energy. The variational problem preserves two important structures, volume conservation and energy dissipation. Then we propose a structure-preserving parametric finite element method (SP-PFEM) to discretize the symmetrized variational problem in space via PFEM and in time via an implicit-explicit Euler method, which preserves the volume in the discretized level. Under a relatively mild and simple condition on \(\gamma (\boldsymbol{n})\) , we show that the SP-PFEM is unconditionally energy stable for almost all anisotropic surface energies \(\gamma (\boldsymbol{n})\) arising in practical applications. Thus the proposed SP-PFEM preserves the two important structures of the original anisotropic surface diffusion in the discretized level. Extensive numerical results are reported to demonstrate the efficiency and accuracy as well as the structure-preserving properties of the proposed SP-PFEM for solving anisotropic surface diffusion in 3D.Keywordsanisotropic surface diffusionCahn–Hoffman \(\xi\) -vectoranisotropic surface energyparametric finite element methodstructure-preservingsurface energy matrixenergy-stableMSC codes65M6065M1235K5553C44

Topics & Concepts