### 2009-23

by Deckelnick, K., Schieweck, F..

**Series:** 2009-23, Preprints

- MSC:
- 35K60 Nonlinear initial value problems for linear parabolic equations
- 65M15 Error bounds
- 65M60 Finite elements, ~Rayleigh-Ritz and Galerkin methods, finite methods

**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 radial variable. 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 C1-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.

**Keywords:**

Willmore flow, Dirichlet boundary conditions, finite elements, error estimates