A SYMMETRIZED PARAMETRIC FINITE ELEMENT METHOD FOR ANISOTROPIC SURFACE DIFFUSION OF CLOSED CURVES

We deal with a long-standing problem about how to design an energy-stable numeri-cal scheme for solving the motion of a closed curve under anisotropic surface diffusion with a general anisotropic surface energy \gamma(n) in two dimensions, where n is the outward unit normal vector. By introducing a novel surface energy matrix Zk(n) which depends on the Cahn-Hoffman-vector and a stabilizing function k(n), we first reformulate the equation into a conservative form and derive a new symmetrized variational formulation for anisotropic surface diffusion with weakly or strongly anisotropic surface energies. Then, a semidiscretization in space for the variational formulation is proposed, and its area conservation and energy dissipation properties are proved. The semidiscretiza-tion is further discretized in time by an implicit structural-preserving scheme (SP-PFEM) which can rigorously preserve the enclosed area in the fully discrete level. Furthermore, we prove that the SP-PFEM is unconditionally energy-stable for almost any anisotropic surface energy \gamma(n) under a simple and mild condition on \gamma(n). For several commonly used anisotropic surface energies, we construct Zk(n) explicitly. Finally, extensive numerical results are reported to demonstrate the high performance of the proposed scheme.