A symmetrized parametric finite element method for anisotropic surface diffusion ii. three dimensions
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For the evolution of a closed surface under anisotropic surface diffusion with a general anisotropic surface energy γ(n) in three dimensions (3D), where n is the unit outward normal vector, by introducing a novel symmetric positive definite surface energy matrix Z_k(n) depending on a stabilizing function k(n) and the Cahn-Hoffman ξ-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 γ(n), we show that SP-PFEM is unconditionally energy-stable for almost all anisotropic surface energies γ(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.
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