A dual formulation and finite element method is proposed and analyzed for simulating the Stefan problem with surface tension. The method uses a mixed form of the heat equation in the solid and liquid (bulk) domains, and imposes a weak formulation of the interface motion law (on the solid liquid interface) as a constraint. The basic unknowns are the heat fluxes and temperatures in the bulk, and the velocity and temperature on the interface. The formulation, as well as its discretization, is viewed as a saddle point system. Well-posedness of the time semi-discrete and fully discrete formulations is proved in three dimensions, as well as an a priori (stability) bound and conservation law. Simulations of interface growth (in two dimensions) are presented to illustrate the method.
Classification :
65-XX, 35-XX
Mots-clés :
Stefan problem, mixed method, energy stability, interface motion, semi-implicit scheme
Affiliations des auteurs :
Christopher B. Davis
1
;
Shawn W. Walker
2
1
Tennessee Tech University, Cookeville, USA
2
Louisiana State University, Baton Rouge, USA
Christopher B. Davis; Shawn W. Walker. A mixed formulation of the Stefan problem with surface tension. Interfaces and free boundaries, Tome 17 (2015) no. 4, pp. 427-464. doi: 10.4171/ifb/349
@article{10_4171_ifb_349,
author = {Christopher B. Davis and Shawn W. Walker},
title = {A mixed formulation of the {Stefan} problem with surface tension},
journal = {Interfaces and free boundaries},
pages = {427--464},
year = {2015},
volume = {17},
number = {4},
doi = {10.4171/ifb/349},
url = {http://geodesic.mathdoc.fr/articles/10.4171/ifb/349/}
}
TY - JOUR
AU - Christopher B. Davis
AU - Shawn W. Walker
TI - A mixed formulation of the Stefan problem with surface tension
JO - Interfaces and free boundaries
PY - 2015
SP - 427
EP - 464
VL - 17
IS - 4
UR - http://geodesic.mathdoc.fr/articles/10.4171/ifb/349/
DO - 10.4171/ifb/349
ID - 10_4171_ifb_349
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%A Shawn W. Walker
%T A mixed formulation of the Stefan problem with surface tension
%J Interfaces and free boundaries
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%P 427-464
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%U http://geodesic.mathdoc.fr/articles/10.4171/ifb/349/
%R 10.4171/ifb/349
%F 10_4171_ifb_349
