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