Numerical approximation of the Steiner problem in dimension 2 and 3
Matthieu Bonnivard, Élie Bretin, Antoine Lemenant · Mathematics of Computation · 2019
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.