A structure-preserving parametric finite element method for geometric flows with anisotropic surface energy

We propose and analyze structure-preserving parametric finite element methods (SP-PFEM) for evolution of a closed curve under different geometric flows with arbitrary anisotropic surface energy $\gamma(\boldsymbol{n})$ for $\boldsymbol{n}\in \mathbb{S}^1$ representing the outward unit normal vector. By introducing a novel surface energy matrix $\boldsymbol{G}_k(\boldsymbol{n})$ depending on $\gamma(\boldsymbol{n})$ and the Cahn-Hoffman $\boldsymbol{\xi}$-vector as well as a nonnegative stabilizing function $k(\boldsymbol{n}):\ \mathbb{S}^1\to \mathbb{R}$, which is a sum of a symmetric positive definite matrix and an anti-symmetric matrix, we obtain a new geometric partial differential equation and its corresponding variational formulation for the evolution of a closed curve under anisotropic surface diffusion. Based on the new weak formulation, we propose a parametric finite element method for the anisotropic surface diffusion and show that it is area conservation and energy dissipation under a very mild condition on $\gamma(\boldsymbol{n})$. The SP-PFEM is then extended to simulate evolution of a close curve under other anisotropic geometric flows including anisotropic curvature flow and area-conserved anisotropic curvature flow. Extensive numerical results are reported to demonstrate the efficiency and unconditional energy stability as well as good mesh quality property of the proposed SP-PFEM for simulating anisotropic geometric flows.