Error analysis for the elastic flow of parametrized curves

Deckelnick, Klaus; Dziuk, Gerhard

doi:10.1090/S0025-5718-08-02176-5

Error analysis for the elastic flow of parametrized curves
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by Klaus Deckelnick and Gerhard Dziuk;

Math. Comp. 78 (2009), 645-671

DOI: https://doi.org/10.1090/S0025-5718-08-02176-5

Published electronically: October 17, 2008

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Abstract:

We analyze a semidiscrete numerical scheme for approximating the evolution of parametric curves by elastic flow in $\mathbb {R}^n$. The fourth order equation is split into two coupled second order problems, which are approximated by linear finite elements. We prove error bounds for the resulting scheme and present numerical test calculations that confirm our analysis.

References

John W. Barrett, Harald Garcke, and Robert Nürnberg, A parametric finite element method for fourth order geometric evolution equations, J. Comput. Phys. 222 (2007), no. 1, 441–462. MR 2298053, DOI 10.1016/j.jcp.2006.07.026
Klaus Deckelnick and Gerhard Dziuk, Error analysis of a finite element method for the Willmore flow of graphs, Interfaces Free Bound. 8 (2006), no. 1, 21–46. MR 2231251, DOI 10.4171/IFB/134
Klaus Deckelnick, Gerhard Dziuk, and Charles M. Elliott, Computation of geometric partial differential equations and mean curvature flow, Acta Numer. 14 (2005), 139–232. MR 2168343, DOI 10.1017/S0962492904000224
G. Dziuk, Computational parametric Willmore flow, Preprint Fakultät für Mathematik und Physik 07–13 (2007).
Gerhard Dziuk, Ernst Kuwert, and Reiner Schätzle, Evolution of elastic curves in $\Bbb R^n$: existence and computation, SIAM J. Math. Anal. 33 (2002), no. 5, 1228–1245. MR 1897710, DOI 10.1137/S0036141001383709
A. Polden, Curves and surfaces of least total curvature and fourth–order flows, Ph.D. dissertation, Universität Tübingen, Tübingen, Germany, 1996.