Error analysis for the approximation of axisymmetric Willmore flow by $C^1$-finite elements

Abstract

We consider the Willmore flow of axially symmetric surfaces subject to Dirichlet boundary conditions. The corresponding evolution is described by a nonlinear parabolic PDE of fourth order for the radius function. A suitable weak form of the equation, which is based on the first variation of the Willmore energy, leads to a semidiscrete scheme, in which we employ piecewise cubic $C^1$-finite elements for the one-dimensional approximation in space.We prove optimal error bounds in Sobolev norms for the solution and its time derivative and present numerical test examples.