Surface Diffusion of Graphs: Variational Formulation, Error Analysis and Simulation

Eberhard Baensch
Weierstrass Institute for Applied Analysis and Stochastics and Freie Universit\"at Berlin
10117 Berlin, Germany.

Pedro Morin
Instituto de Matem\'atica Aplicada del Litoral (IMAL) and Departamento de Matem\'atica
Facultad de Ingenieria Quimica
Universidad Nacional del Litoral
3000 Santa Fe, Argentina.

Ricardo H. Nochetto
Department of Mathematics
University of Maryland, College Park
College Park, MD 20742, USA.


Surface diffusion is a (4th order highly nonlinear) geometric driven motion of a surface with normal velocity proportional to the surface Laplacian of mean curvature. We present a novel variational formulation for graphs and derive a priori error estimates for a time-continuous finite element discretization. We also introduce a semi-implicit time discretization and a Schur complement approach to solve the resulting fully discrete, linear systems. After computational verification of the orders of convergence for polynomial degrees 1 and 2, we show several simulations in 1d and 2d with and without forcing which explore the smoothing effect of surface diffusion as well as the onset of singularities in finite time, such as infinite slopes and cracks.