A symmetrized parametric finite element method for anisotropic surface diffusion of closed curves via a Cahn-Hoffman -vector formulation
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We deal with a long-standing problem about how to design an energy-stable numerical scheme for solving the motion of a closed curve under anisotropic surface diffusion with a general anisotropic surface energy γ(n) in two dimensions, where n is the outward unit normal vector. By introducing a novel symmetric positive definite surface energy matrix Z_k(n) depending on the Cahn-Hoffman ξ-vector and a stabilizing function k(n), we first reformulate the anisotropic surface diffusion into a conservative form and then derive a new symmetrized variational formulation for the anisotropic surface diffusion with weakly or strongly anisotropic surface energies. A semi-discretization in space for the symmetrized variational formulation is proposed and its area (or mass) conservation and energy dissipation are proved. The semi-discretization is then discretized in time by either an implicit structural-preserving scheme (SP-PFEM) which preserves the area in the discretized level or a semi-implicit energy-stable method (ES-PFEM) which needs only solve a linear system at each time step. Under a relatively simple and mild condition on γ(n), we show that both SP-PFEM and ES-PFEM are unconditionally energy-stable for almost all anisotropic surface energies γ(n) arising in practical applications. Specifically, for several commonly-used anisotropic surface energies, we construct Z_k(n) explicitly. Finally, extensive numerical results are reported to demonstrate the high performance of the proposed numerical schemes.
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