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Curve shortening flow coupled to lateral diffusion
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Pozzi, Paola and Stinner, Björn (2017) Curve shortening flow coupled to lateral diffusion. Numerische Mathematik, 135 (4). pp. 1171-1205. doi:10.1007/s00211-016-0828-8 ISSN 0029-599X.
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Official URL: http://doi.org/ 10.1007/s00211-016-0828-8
Abstract
We present and analyze a semi-discrete finite element scheme for a system consisting of a geometric evolution equation for a curve and a parabolic equation on the evolving curve. More precisely, curve shortening flow with a forcing term that depends on a field defined on the curve is coupled with a diffusion equation for that field. The scheme is based on ideas of [Dziuk, G. Discrete anisotropic curve shortening flow, SIAM J. Numer. Anal. 36, 6 (1999), 1808–1830] for the curve shortening flow and [Dziuk, G., and Elliott, C. M. Finite elements on evolving surfaces, IMA J. Numer. Anal. 27, 2 (2007), 262–292] for the parabolic equation on the moving curve. Additional estimates are required in order to show convergence, most notably with respect to the length element: While in [Dziuk, G. Discrete anisotropic curve shortening flow, SIAM J. Numer. Anal. 36, 6 (1999), 1808–1830] an estimate of its error was sufficient we here also need to estimate the time derivative of the error which arises from the diffusion equation. Numerical simulation results support the theoretical findings.
| Item Type: | Journal Article | ||||||||
|---|---|---|---|---|---|---|---|---|---|
| Subjects: | Q Science > QA Mathematics | ||||||||
| Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||||||
| Library of Congress Subject Headings (LCSH): | Discrete element method, Finite element method, Evolution equations, Differential equations, Parabolic | ||||||||
| Journal or Publication Title: | Numerische Mathematik | ||||||||
| Publisher: | Springer | ||||||||
| ISSN: | 0029-599X | ||||||||
| Official Date: | April 2017 | ||||||||
| Dates: |
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| Volume: | 135 | ||||||||
| Number: | 4 | ||||||||
| Page Range: | pp. 1171-1205 | ||||||||
| DOI: | 10.1007/s00211-016-0828-8 | ||||||||
| Status: | Peer Reviewed | ||||||||
| Publication Status: | Published | ||||||||
| Access rights to Published version: | Restricted or Subscription Access | ||||||||
| Date of first compliant deposit: | 19 July 2016 | ||||||||
| Date of first compliant Open Access: | 5 August 2017 | ||||||||
| Funder: | Isaac Newton Institute for Mathematical Sciences | ||||||||
| Related URLs: |
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