2022/11/01 by Bao, Weizhu, Li, Yifei · 2 citations
#35K55 #53C44 #65M12 #65M60 #FOS: Mathematics #Numerical Analysis (math.NA)
paper · doi:10.48550/arxiv.2211.00297
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 γ(\boldsymboln) for \boldsymboln∈ \mathbbS1 representing the outward unit normal vector. By introducing a novel surface energy matrix \boldsymbolGk(\boldsymboln) depending on γ(\boldsymboln) and the Cahn-Hoffman \boldsymbolξ-vector as well as a nonnegative stabilizing function k(\boldsymboln): \mathbbS1→ 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 γ(\boldsymboln). 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.