A Symmetrized Parametric Finite Element Method for Anisotropic Surface Diffusion of Closed Curves

Abstract

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 in two dimensions, where is the outward unit normal vector. By introducing a novel surface energy matrix which depends on the Cahn–Hoffman -vector and a stabilizing function , 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 semidiscretization 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 under a simple and mild condition on . For several commonly used anisotropic surface energies, we construct explicitly. Finally, extensive numerical results are reported to demonstrate the high performance of the proposed scheme.