Theoretical and numerical study of a free boundary problem by boundary integral methods
Crouzeix, Michel1 ; Féat, Philippe ; Sayas, Francisco-Javier2
1 Université de Rennes 1 IRMAR, UMR 6625 Campus de Beaulieu 35042 Rennes Cedex FRANCE
2 Dep. Matemática Aplicada, Universidad de Zaragoza, Centro Politécnico Superior, c/ María de Luna, 3–50015 Zaragoza, Spain.
ESAIM: Modélisation mathématique et analyse numérique, Tome 35 (2001) no. 6, pp. 1137-1158.

Dans cet article on considère un problème à frontière libre intervenant en formage électromagnétique. Après l'avoir ramené à un système intégro-différentiel où l'inconnue est la représentation paramétrique de la frontière, on en étudie les propriétés mathématiques essentielles. On s'intéresse ensuite à l'approximation numérique par des méthodes de type Galerkin ou de collocation en utilisant pour l'approximation des polynômes trigonométriques ou des fonctions splines.

In this paper we study a free boundary problem appearing in electromagnetism and its numerical approximation by means of boundary integral methods. Once the problem is written in a equivalent integro-differential form, with the arc parametrization of the boundary as unknown, we analyse it in this new setting. Then we consider Galerkin and collocation methods with trigonometric polynomial and spline curves as approximate solutions.

Classification : 35R35, 41A15, 42A12, 45G05, 65R20
Mots-clés : free boundary, spline, trigonometric polynomial
Crouzeix, Michel 1 ; Féat, Philippe  ; Sayas, Francisco-Javier 2

1 Université de Rennes 1 IRMAR, UMR 6625 Campus de Beaulieu 35042 Rennes Cedex FRANCE
2 Dep. Matemática Aplicada, Universidad de Zaragoza, Centro Politécnico Superior, c/ María de Luna, 3–50015 Zaragoza, Spain. 
@article{M2AN_2001__35_6_1137_0,
     author = {Crouzeix, Michel and F\'eat, Philippe and Sayas, Francisco-Javier},
     title = {Theoretical and numerical study of a free boundary problem by boundary integral methods},
     journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
     pages = {1137--1158},
     publisher = {EDP-Sciences},
     volume = {35},
     number = {6},
     year = {2001},
     mrnumber = {1873520},
     zbl = {0985.35114},
     language = {en},
     url = {http://www.numdam.org/item/M2AN_2001__35_6_1137_0/}
}
TY  - JOUR
AU  - Crouzeix, Michel
AU  - Féat, Philippe
AU  - Sayas, Francisco-Javier
TI  - Theoretical and numerical study of a free boundary problem by boundary integral methods
JO  - ESAIM: Modélisation mathématique et analyse numérique
PY  - 2001
SP  - 1137
EP  - 1158
VL  - 35
IS  - 6
PB  - EDP-Sciences
UR  - http://www.numdam.org/item/M2AN_2001__35_6_1137_0/
LA  - en
ID  - M2AN_2001__35_6_1137_0
ER  - 
%0 Journal Article
%A Crouzeix, Michel
%A Féat, Philippe
%A Sayas, Francisco-Javier
%T Theoretical and numerical study of a free boundary problem by boundary integral methods
%J ESAIM: Modélisation mathématique et analyse numérique
%D 2001
%P 1137-1158
%V 35
%N 6
%I EDP-Sciences
%U http://www.numdam.org/item/M2AN_2001__35_6_1137_0/
%G en
%F M2AN_2001__35_6_1137_0
Crouzeix, Michel; Féat, Philippe; Sayas, Francisco-Javier. Theoretical and numerical study of a free boundary problem by boundary integral methods. ESAIM: Modélisation mathématique et analyse numérique, Tome 35 (2001) no. 6, pp. 1137-1158. http://www.numdam.org/item/M2AN_2001__35_6_1137_0/

[1] H.W. Alt and L.A. Caffarelli, Existence and regularity for a minimum problem with a free boundary. J. Reine Angew. Math. 25 (1981) 105-144. | Zbl

[2] O. Coulaud and A. Henrot, Numerical approximation of a free boundary problem arising in electromagnetic shaping. SIAM J. Numer. Anal. 31 (1994) 1109-1127. | Zbl

[3] M. Crouzeix, Variational approach of magnetic shaping problem. Eur. J. Mech. B/Fluids 10 (1991) 627-536. | Zbl

[4] J. Descloux, Stability of solutions of the bidimensional magnetic shaping problem in absence of surface tension. Eur. J. Mech. B/Fluids 10 (1991) 513-526. | Zbl

[5] Ph. Féat, Approximation d'un problème de frontière libre bidimensionnel. Thèse de l'Université de Rennes I, France (1998).

[6] A. Friedman, Variational Principles and Free Boundary Problems. John Wiley & Sons, New York (1982). | MR | Zbl

[7] B. Gustafsson and H. Shagholian, Existence and geometric properties of solutions of a free boundary problem in potential theory. J. Reine Angew. Math. 68 (1996) 137-179. | Zbl

[8] A. Henrot, Subsolutions and supersolutions in a free boundary problem. Ark. Mat. 32 (1994) 79-98. | Zbl

[9] A. Henrot and M. Pierre, Un problème inverse en formage des métaux liquides. RAIRO Modél. Math. Anal. Numér. 23 (1989) 155-177. | Numdam | Zbl

[10] R. Kress, Linear Integral Equations. Springer, New York (1989). | MR | Zbl

[11] W. Mclean and W.L. Wendland, Trigonometric approximation of solutions of periodic pseudodifferential equations. Oper. Theory: Adv. Appl. 41 (1989) 359-383. | Zbl

[12] S. Mikhlin and S. Prößdorf, Singular Integral Operators. Springer-Verlag, Berlin (1986). | MR | Zbl

[13] X. Pelgrin, Un problème de frontière libre. Thèse de l'Université de Rennes I, France (1994).

[14] M. Pierre and J.R. Roche, Numerical simulation of tridimensional electromagnetic shaping of liquid metals. Numer. Math. 65 (1993) 203-217. | EuDML | Zbl

[15] S. Prößdorf and B. Silbermann, Numerical Analysis for Integral and Related Operator Equations. Akademie-Verlag, Berlin (1991). | MR | Zbl

[16] J. Saranen and L. Schroderus, Quadrature methods for strongly elliptic equations of negative order on smooth closed curves. SIAM J. Numer. Anal. 30 (1993) 1769-1795. | Zbl

[17] Y. Yan and I.H. Sloan, On integral equations of the first kind with logarithmic kernels. J. Integral Equations. Appl. 1 (1988) 549-579. | Zbl