Numerical analysis of a steepest-descent pde model for surface relaxation below the roughening temperature

We study the numerical solution of a PDE describing the relaxation of a crystal surface to a flat facet. The PDE is a singular, nonlinear, fourth order evolution equation, which can be viewed as the gradient flow of a convex but nonsmooth energy with respect to the H _per ^-1 inner product. Our numerical scheme uses implicit discretization in time and a mixed finite element approximation in space. The singular character of the energy is handled using regularization, combined with a primal-dual method. We study the convergence of this scheme, both theoretically and numerically.