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 \(\gamma ({\boldsymbol{n}})\) in two dimensions, where \({\boldsymbol{n}}\) is the outward unit normal vector. By introducing a novel surface energy matrix \(\boldsymbol{Z}_k({\boldsymbol{n}})\) which depends on the Cahn–Hoffman \(\boldsymbol{\xi }\) -vector and a stabilizing function \(k({\boldsymbol{n}})\) , 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 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 \(\gamma ({\boldsymbol{n}})\) under a simple and mild condition on \(\gamma ({\boldsymbol{n}})\) . For several commonly used anisotropic surface energies, we construct \(\boldsymbol{Z}_k({\boldsymbol{n}})\) explicitly. Finally, extensive numerical results are reported to demonstrate the high performance of the proposed scheme.Keywordsanisotropic surface diffusionCahn–Hoffman \(\boldsymbol{\xi }\) -vectoranisotropic surface energyparametric finite element methodstructure-preservingenergy-stablesurface energy matrixMSC codes65M6065M1235K5553C44

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