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2009-23

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

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