  We propose an energy-stable parametric finite element method (ES-PFEM) for
simulating solid-state dewetting of thin films in two dimensions via a
sharp-interface model, which is governed by surface diffusion and contact line
(point) migration together with proper boundary conditions. By reformulating
the relaxed contact angle condition into a Robin-type boundary condition and
then treating it as a natural boundary condition, we obtain a new variational
formulation for the problem, in which the interface curve and its contact
points are evolved simultaneously. Then, the variational problem is discretized
in space by using piecewise linear elements. A full discretization is presented
by adopting the backward Euler method in time, and the well-posedness and
energy dissipation of the full discretization are established. The numerical
method is semi-implicit (i.e., a linear system to be solved at each time step
and thus efficient), unconditionally energy-stable with respect to the time
step, and second-order in space measured by a manifold distance between two
curves. In addition, it demonstrates equal mesh distribution when the solution
reaches its equilibrium, i.e., long-time dynamics. Numerical results are
reported to show accuracy and efficiency as well as some good properties of the
proposed numerical method.
