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.
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Klaus Deckelnick, Friedhelm Schieweck, Error analysis for the approximation of axisymmetric Willmore flow by <i>C</i><sup>1</sup>-finite elements. Interfaces Free Bound. 12 (2010), no. 4, pp. 551–574