Finite element-based level set methods for higher order flows

In this paper we shall discuss the numerical simulation of geometric flows by level set methods. Main examples under considerations are higher order flows, such as surface diffusion and Willmore flow as well as variants of them with more complicated surface energies. Such problems find various applications, e.g. in materials science (thin film growth, grain boundary motion), biophysics (membrane shapes), and computer graphics (surface smoothing and restoration). We shall use spatial discretizations by finite element methods and semi-implicit time stepping based on local variational principles, which allows to maintain dissipation properties of the flows by the discretization. In order to compensate for the missing maximum principle, which is indeed a major hurdle for the application of level set methods to higher order flows, we employ frequent redistancing of the level set function. Finally we also discuss the solution of the arising discretized linear systems in each time step and some particular advantages of the finite element approach such as the variational formulation which allows to handle the higher order and various anisotropies efficiently and the possibility of local adaptivity around the zero level set.