A perimeter-decreasing and area-conserving algorithm for surface diffusion flow of curves

A fully discrete finite element method, based on a new weak formulation and time-stepping scheme, is proposed for the surface diffusion flow of closed curves in two-dimensional plane. It proved that method can preserve two geometric structures simultaneously level, i.e., perimeter curve decreases time while area enclosed by conserved. Numerical examples are provided to demonstrate convergence effectiveness preserving structures.