Finite Element-Based Level Set Methods for Higher Order Flows

In this paper we shall discuss the numerical simulation of higher order geometric flows by level set methods. Main examples under considerations are surface diffusion and the Willmore flow as well as variants and of them with more complicated surface energies. Such problems find various applications, e.g. in materials science (crystal growth, thin film technology), 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. We shall review suitable schemes used for redistancing in two and three spatial dimensions. 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 possibility of local adaptivity around the zero level set.