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

Abstract

Abstract: 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.

Keywords: Surface diffusion, fourth-order parabolic problem, finite elements, a priori error estimates, Schur complement, smoothing effect.

AMS Subject Classifications: 35K55, 65M12, 65M15, 65M60, 65Z05.


Published: SIAM Journal on Numerical Analysis, Volume 42, Number 2 (2004), 773--799.