Finite Element Methods for Surface Diffusion

   Ricardo Nochetto

  University of Maryland

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 which is
semi-implicit, requires no explicit parametrization, and yields a
linear system of lower order elliptic PDE to solve at each time
step. We develop a finite element method, and propose a Schur
complement approach to solve the resulting linear systems. We
also introduce a new graph formulation and show stability and
optimal a priori error estimates for position and curvature. We
illustrate both approaches with several simulations of curves and
surfaces, some exhibiting singularites (such as pinch-off and
crack formation) in finite time. The beneficial use of mesh
smoothing, which helps prevent mesh distortion, and space-time
adaptivity is also addressed.