A structure-preserving parametric finite element method for geometric flows with anisotropic surface energy
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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(n) for n∈𝕊^1 representing the outward unit normal vector. By introducing a novel surface energy matrix G_k(n) depending on γ(n) and the Cahn-Hoffman ξ-vector as well as a nonnegative stabilizing function k(n): 𝕊^1→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(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.
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