A $C^1$–finite element method for the Willmore flow of two-dimensional graphs
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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.References
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Additional Information
- Klaus Deckelnick
- Affiliation: Institut für Analysis und Numerik, Otto–von–Guericke–Universität Magdeburg, Universitätsplatz 2, 39106 Magdeburg, Germany
- MR Author ID: 318167
- 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
- MR Author ID: 155960
- Email: schiewec@ovgu.de
- 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.
- © Copyright 2015 American Mathematical Society
- Journal: Math. Comp. 84 (2015), 2617-2643
- MSC (2010): Primary 65M15, 65M60; Secondary 35K59
- DOI: https://doi.org/10.1090/mcom/2973
- MathSciNet review: 3378841