A finite element method for surface diffusion: the parametric case

Pedro Morin, Eberhard Bänsch, Ricardo H. Nochetto


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 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.

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

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

Published: Journal of Computational Physics 203 (2005) 321--343.

Published: Journal of Computational Physics 203 (2005) 321--343.

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