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Error analysis for the approximation of axisymmetric Willmore flow by C1-elements

by Deckelnick, K., Schieweck, F..

Series: 2009-23, Preprints

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

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.

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