Finite Element Methods for Surface Diffusion

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 the parametric case, 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 state an optimal a priori error estimate. We conclude with several significant simulations, some with pinch-off in finite time.

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

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


Published: Free Boundary Problems. International Series of Num. Math., vol. 147, 53--63, Birkhäuser (2003)