Numerical Analysis of a Steepest-Descent PDE Model for Surface Relaxation below the Roughening Temperature
Robert V. Kohn, Henrique M. Versieux · SIAM Journal on Numerical Analysis · 2010
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.