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A $ C^1$-finite element method for the Willmore flow of two-dimensional graphs


Authors: Klaus Deckelnick, Jakob Katz and Friedhelm Schieweck
Journal: Math. Comp. 84 (2015), 2617-2643
MSC (2010): Primary 65M15, 65M60; Secondary 35K59
DOI: https://doi.org/10.1090/mcom/2973
Published electronically: May 12, 2015
MathSciNet review: 3378841
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Abstract: 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 $ C^1$-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.


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Additional Information

Klaus Deckelnick
Affiliation: Institut für Analysis und Numerik, Otto–von–Guericke–Universität Magdeburg, Universitätsplatz 2, 39106 Magdeburg, Germany
Email: klaus.deckelnick@ovgu.de

Jakob Katz
Affiliation: Institut für Analysis und Numerik, Otto–von–Guericke–Universität Magdeburg, Universitätsplatz 2, 39106 Magdeburg, Germany
Email: jakob.katz@st.ovgu.de

Friedhelm Schieweck
Affiliation: Institut für Analysis und Numerik, Otto–von–Guericke–Universität Magdeburg, Universitätsplatz 2, 39106 Magdeburg, Germany
Email: schiewec@ovgu.de

DOI: https://doi.org/10.1090/mcom/2973
Keywords: Willmore flow, Dirichlet boundary conditions, $C^1$-finite elements, error estimates
Received by editor(s): February 15, 2013
Received by editor(s) in revised form: January 13, 2014
Published electronically: May 12, 2015
Additional Notes: The authors gratefully acknowledge financial support from the German Research Council (DFG) through grant DE 611/5-2.
Article copyright: © Copyright 2015 American Mathematical Society