A finite element method for surface diffusion: the parametric case

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 parametric surfaces with or without boundaries. The method is semi-implicit, requires no explicit parametrization, and yields a linear system of elliptic PDE to solve at each time step. We next develop a finite element method, propose a Schur complement approach to solve the resulting linear systems, and show several significant simulations, some with pinch-off in finite time. We introduce a mesh regularization algorithm, which helps prevent mesh distortion, and discuss the use of time and space adaptivity to increase accuracy while reducing complexity.