An energy-stable parametric finite element method for anisotropic surface diffusion

We propose an energy-stable parametric finite element method (ES-PFEM) to discretize the motion of a closed curve under surface diffusion with anisotropic energy $\gamma(\theta)$ -- in two dimensions, while $\theta$ is angle between outward unit normal vector and vertical axis. By introducing positive definite (density) matrix $G(\theta)$, we present new simple variational formulation for prove that it satisfies area/mass conservation dissipation. The problem discretized space by dissipation are established semi-discretization. Then further time (semi-implicit) backward Euler so only linear system be solved at each step full-discretization thus efficient. establish well-posedness identify some conditions on such keeps unconditionally energy-stable. Finally ES-PFEM applied simulate solid-state dewetting thin films energies, i.e. open proper boundary triple points moving along horizontal substrate. Numerical results reported demonstrate efficiency accuracy as well proposed ES-PFEM.