A symmetrized parametric finite element method for anisotropic surface diffusion ii. three dimensions

Abstract

. 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.