Speaker
Description
Mean curvature flow evolves a family of hypersurfaces in such a way that the normal velocity is given by the mean curvature at each point of the surface. This fundamental geometric evolution law can be interpreted as the $L^2$-gradient flow of the area functional and has applications e.g. in materials science and image processing. The talk is concerned with the analysis of a numerical method for the approximation of mean curvature flow in a parametric setting. Our starting point is an approach suggested by Elliott and Fritz that derives a parabolic system for the position vector of the evolving hypersurfaces using a reparametrization via the DeTurck trick. This technique introduces a tangential velocity that leads to strict parabolicity of the system. Based on a natural weak formulation we introduce a finite element semidiscretization with continuous, piecewise polynomial elements of order $k \geq 2$ that are defined on a fixed reference hypersurface. As our main result we shall present an error analysis for the resulting scheme that yields in particular an optimal $H^1$-error bound for the position vector. In addition, we show a couple of numerical results in order to illustrate that the tangential motion leads to good mesh properties. This is work obtained in collaboration with Vanessa Styles (University of Sussex).