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Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 
 

 

Numerical approximation of the Steiner problem in dimension $ 2$ and $ 3$


Authors: Matthieu Bonnivard, Elie Bretin and Antoine Lemenant
Journal: Math. Comp. 89 (2020), 1-43
MSC (2010): Primary 49M25, 65M12
DOI: https://doi.org/10.1090/mcom/3442
Published electronically: May 14, 2019
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Abstract: The aim of this work is to present some numerical computations of solutions of the Steiner problem, based on the recent phase field approximations proposed by Lemenant and Santambrogio [C. R. Math. Acad. Sci. Paris 352 (2014), 451-454] and analyzed by Bonnivard, Lemenant, Millot, and Santambrogio. Our strategy consists in improving the regularity of the associated phase field solution by use of higher-order derivatives in the Cahn-Hilliard functional as in [Burger, Esposito, and Zeppieri, Multiscale Model Simul. 13 (2015), 1354-1389]. We justify the convergence of this slightly modified version of the functional, together with other techniques that we employ to improve the numerical experiments. In particular, we are able to consider a large number of points in dimension 2. We finally present and justify an approximation method that is efficient in dimension 3, which is one of the major novelties of the paper.


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Additional Information

Matthieu Bonnivard
Affiliation: Université Paris-Diderot, Sorbonne Paris-Cité, Sorbonne Université, CNRS, Laboratoire Jacques-Louis Lions, F-75013 Paris, France
Email: bonnivard@ljll.univ-paris-diderot.fr

Elie Bretin
Affiliation: Université Lyon, INSA de Lyon, CNRS UMR 5208, Institut Camille Jordan, 20 avenue Albert Einstein, F-69621 Villeurbanne Cedex, France
Email: elie.bretin@insa-lyon.fr

Antoine Lemenant
Affiliation: Université Paris-Diderot, Sorbonne Paris-Cité, Sorbonne Université, CNRS, Laboratoire Jacques-Louis Lions, F-75013 Paris, France
Email: lemenant@ljll.univ-paris-diderot.fr

DOI: https://doi.org/10.1090/mcom/3442
Received by editor(s): April 10, 2018
Received by editor(s) in revised form: February 13, 2019
Published electronically: May 14, 2019
Additional Notes: This work was supported by the PGMO project COCA from the Hadamard foundation.
Article copyright: © Copyright 2019 American Mathematical Society