Stability of a simple scheme for the approximation of elastic knots and self-avoiding inextensible curves
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- by Sören Bartels and Philipp Reiter;
- Math. Comp. 90 (2021), 1499-1526
- DOI: https://doi.org/10.1090/mcom/3633
- Published electronically: May 6, 2021
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Abstract:
We discuss a semi-implicit numerical scheme that allows for minimizing the bending energy of curves within certain isotopy classes. To this end we consider a weighted sum of the bending energy and the tangent-point functional.
Based on estimates for the second derivative of the latter and a uniform bi-Lipschitz radius, we prove a stability result implying energy decay during the evolution as well as maintenance of arclength parametrization.
Finally we present some numerical experiments exploring the energy landscape, targeted to the question how to obtain global minimizers of the bending energy in knot classes, so-called elastic knots.
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Bibliographic Information
- Sören Bartels
- Affiliation: Abteilung für Angewandte Mathematik, Albert-Ludwigs-Universität Freiburg, Hermann-Herder-Str. 10, 79104 Freiburg i. Br., Germany
- Email: bartels@mathematik.uni-freiburg.de
- Philipp Reiter
- Affiliation: Faculty of Mathematics, Chemnitz University of Technology, 09107 Chemnitz, Germany
- MR Author ID: 854824
- ORCID: 0000-0001-9651-9445
- Email: philipp.reiter@mathematik.tu-chemnitz.de
- Received by editor(s): July 12, 2019
- Received by editor(s) in revised form: November 12, 2020
- Published electronically: May 6, 2021
- Additional Notes: The second author was partially supported by DFG-Grant RE 3930/1–1.
- © Copyright 2021 American Mathematical Society
- Journal: Math. Comp. 90 (2021), 1499-1526
- MSC (2020): Primary 65N12; Secondary 65N15, 65N30
- DOI: https://doi.org/10.1090/mcom/3633
- MathSciNet review: 4273107