Elliott, Charles M. and Fritz, Hans (2017) On approximations of the curve shortening flow and of the mean curvature flow based on the DeTurck trick. IMA JOURNAL OF NUMERICAL ANALYSIS, 37 (2). pp. 543-603. ISSN 0272-4979, 1464-3642
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In this article we discuss novel numerical schemes for the computation of the curve shortening and mean curvature flows that are based on special reparametrizations. The main idea is to use special solutions to the harmonic map heat flow in order to reparametrize the equations of motion. This idea is widely known from the Ricci flow as the De Turck trick. By introducing a variable timescale for the harmonic map heat flow, we obtain families of numerical schemes for the reparametrized flows. For the curve shortening flow this family unveils a surprising geometric connection between the numerical schemes in Barrett et al. (2011, Numer. Methods Partial Differential Equations, 27, 1-30) and Deckelnick & Dziuk (1994, On the approximation of the curve shortening flow. Calculus of Variations, Applications and Computations: Pont-a-Mousson, Pitman Research Notes in Mathematics Series, pp. 100-108). For the mean curvature flow we obtain families of schemes with good mesh properties similar to those in Barrett et al. (2008a, J. Comput. Phys., 227, 4281-4307). We prove error estimates for the semidiscrete scheme of the curve shortening flow. The behaviour of the fully discrete schemes with respect to the redistribution of mesh points is studied in numerical experiments. We also discuss possible generalizations of our ideas to other extrinsic flows.
| Item Type: | Article |
|---|---|
| Uncontrolled Keywords: | GEOMETRIC EVOLUTION-EQUATIONS; PARAMETRIC APPROXIMATION; HARMONIC MAPS; PLANE-CURVES; SURFACES; curve shortening flow; mean curvature flow; harmonic map heat flow; DeTurck trick; parametric finite elements; error estimates; tangential redistributions; mesh properties |
| Subjects: | 500 Science > 510 Mathematics |
| Divisions: | Mathematics |
| Depositing User: | Dr. Gernot Deinzer |
| Date Deposited: | 14 Dec 2018 13:11 |
| Last Modified: | 20 Feb 2019 14:57 |
| URI: | https://pred.uni-regensburg.de/id/eprint/1091 |
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