  We proposed a structure-preserving stabilized parametric finite element
method (SPFEM) for the evolution of closed curves under anisotropic surface
diffusion with an arbitrary surface energy $\hat{\gamma}(\theta)$. By
introducing a non-negative stabilizing function $k(\theta)$ depending on
$\hat{\gamma}(\theta)$, we obtained a novel stabilized conservative weak
formulation for the anisotropic surface diffusion. A SPFEM is presented for the
discretization of this weak formulation. We construct a comprehensive framework
to analyze and prove the unconditional energy stability of the SPFEM under a
very mild condition on $\hat{\gamma}(\theta)$. This method can be applied to
simulate solid-state dewetting of thin films with arbitrary surface energies,
which are characterized by anisotropic surface diffusion and contact line
migration. Extensive numerical results are reported to demonstrate the
efficiency, accuracy and structure-preserving properties of the proposed SPFEM
with anisotropic surface energies $\hat{\gamma}(\theta)$ arising from different
applications.
