Preprint series: 09-23, Preprints
- 35K60 Nonlinear boundary value problems for linear parabolic PDE; boundary value problems for nonlinear parabolic PDE
- 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
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