A novel finite element scheme for curve diffusion

We consider the motion of planar closed curves by curve diffusion. The underlying evolution law gives rise to a fourth order, highly nonlinear parabolic problem with the property that the length of the evolving curves decreases in time while the enclosed area remains fixed. We propose a new finite element scheme that is based on a parametric description of the curves and includes a specifically chosen tangential motion which is designed to enhance a parametrisation that is proportional to arclength.
The system is discretised using continuous, piecewise linear finite elements in space and an Euler scheme in time. We report on the numerical analysis of the resulting method and present results of test calculations. This is joint work with Robert Nürnberg.