A two-dimensional Stefan problem is usually introduced as a model of solidification, melting or sublimation phenomena. The two-phase Stefan problem has been studied as a direct problem, where the free boundary separating the two regions is eliminated using a variational inequality (Baiocchi, 1977; Baiocchi et al., 1973; Rodrigues, 1980; Saguez, 1980; Srunk and Friedman, 1994), the enthalpy function (Ciavaldini, 1972; Lions, 1969; Nochetto et al., 1991; Saguez, 1980), or a control problem (El Bagdouri, 1987; Peneau, 1995; Saguez, 1980). In the present work, we provide a new formulation leading to a shape optimization problem. For a semidiscretization in time, we consider an Euler scheme. Under some restrictions related to stability conditions, we prove an L^2 -rate of convergence of order 1 for the temperature. In the last part, we study the existence of an optimal shape, compute the shape gradient, and suggest a numerical algorithm to approximate the free boundary. The numerical results obtained show that this method is more efficient compared with the others.
@article{bwmeta1.element.bwnjournal-article-amcv11i4p921bwm,
author = {Aboula\"\i ch, Rajae and Haggouch, Ilham and Souissi, Ali},
title = {Approximation of a solidification problem},
journal = {International Journal of Applied Mathematics and Computer Science},
volume = {11},
year = {2001},
pages = {921-955},
zbl = {1015.80008},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-amcv11i4p921bwm}
}
Aboulaïch, Rajae; Haggouch, Ilham; Souissi, Ali. Approximation of a solidification problem. International Journal of Applied Mathematics and Computer Science, Tome 11 (2001) pp. 921-955. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-amcv11i4p921bwm/
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