Volume 57, Number 2, March-April 2023
|Page(s)||445 - 466|
|Published online||23 March 2023|
Convergence of a scheme for an elastic flow with tangential mesh movement
Fakultät für Mathematik, Universität Duisburg-Essen, Thea-Leymann-Straße 9, 45127 Essen, Germany
2 Mathematics Institute, Zeeman Building, University of Warwick, Coventry CV4 7AL, UK
* Corresponding author: firstname.lastname@example.org
Accepted: 8 November 2022
Elastic flow for closed curves can involve significant deformations. Mesh-based approximation schemes require tangentially redistributing vertices for long-time computations. We present and analyze a method that uses the Dirichlet energy for this purpose. The approach effectively also penalizes the length of the curve, and equilibrium shapes are equivalent to stationary points of the elastic energy augmented with the length functional. Our numerical method is based on linear parametric finite elements. Following the lines of Deckelnick and Dziuk [Math. Comp. 78 (2009) 645–671] we prove convergence and establish error estimates, noting that the addition of the Dirichlet energy simplifies the analysis in comparison with the length functional. We also present a simple semi-implicit time discretization and discuss some numerical results that support the theory.
Mathematics Subject Classification: 65M15 / 65M20 / 65M60 / 53E40 / 35K65
Key words: Elastic flow / gradient flow / finite element method / semi-discrete scheme
© The authors. Published by EDP Sciences, SMAI 2023
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