  We propose a structure-preserving parametric finite element method (SP-PFEM)
for discretizing the surface diffusion of a closed curve in two dimensions (2D)
or surface in three dimensions (3D). Here the "structure-preserving" refers to
preserving the two fundamental geometric structures of the surface diffusion
flow: (i) the conservation of the area/volume enclosed by the closed
curve/surface, and (ii) the decrease of the perimeter/total surface area of the
curve/surface. For simplicity of notations, we begin with the surface diffusion
of a closed curve in 2D and present a weak (variational) formulation of the
governing equation. Then we discretize the variational formulation by using the
backward Euler method in time and piecewise linear parametric finite elements
in space, with a proper approximation of the unit normal vector by using the
information of the curves at the current and next time step. The constructed
numerical method is shown to preserve the two geometric structures and also
enjoys the good property of asymptotic equal mesh distribution. The proposed
SP-PFEM is "weakly" implicit (or almost semi-implicit) and the nonlinear system
at each time step can be solved very efficiently and accurately by the Newton's
iterative method. The SP-PFEM is then extended to discretize the surface
diffusion of a closed surface in 3D. Extensive numerical results, including
convergence tests, structure-preserving property and asymptotic equal mesh
distribution, are reported to demonstrate the accuracy and efficiency of the
proposed SP-PFEM for simulating surface diffusion in 2D and 3D.
