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A C1-finite element method for the Willmore flow of two-dimensional graphs

by Deckelnick, K., Katz, J., Schieweck, F..

Series: 2013-04, Preprints

65M15 Error bounds
65M60 Finite elements, ~Rayleigh-Ritz and Galerkin methods, finite methods

We consider the Willmore flow of two-dimensional graphs subject to Dirichlet boundary conditions. The corresponding evolution is described by a highly nonlinear parabolic PDE of fourth order for the height function. Based on a suitable weak form of the equation we derive a semidiscrete scheme which uses C1-finite elements and interpolates the Dirichlet boundary conditions. We prove quasioptimal error bounds in Sobolev norms for the solution and its time derivative and present results of test calculations.

Willmore flow, Dirichlet boundary conditions, C1-finite elements, error estimates