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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 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.
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