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

TL;DR

This paper introduces a novel symmetric variational formulation and a structure-preserving finite element method for simulating 3D anisotropic surface diffusion, ensuring volume conservation and energy stability.

Contribution

It develops a new symmetrized variational formulation and a finite element method that preserves volume and energy stability for anisotropic surface diffusion in 3D.

Findings

Method is unconditionally energy-stable for most practical anisotropic energies.

Numerical results confirm the method's efficiency and accuracy.

The approach effectively preserves volume during simulations.

Abstract

For the evolution of a closed surface under anisotropic surface diffusion with a general anisotropic surface energy $\gamma(\boldsymbol{n})$ 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 $\boldsymbol{\xi}$-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 $\gamma(\boldsymbol{n})$, we show that SP-PFEM is unconditionally energy-stable for almost all anisotropic surface energies $\gamma(\boldsymbol{n})$ 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.