A finite element method for surface diffusion: The parametric case
- Bänsch, Eberhard
- Morin, Pedro
- Nochetto, Ricardo H.
2010 Mathematics Subject Classification
- 35K55 65M12 65M15 65M60 65Z05
- Surface diffusion, fourth-order parabolic problem, finite elements, Schur complement, smoothing effect, pinch-off
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.