Crouzeix, Michel; Féat, Philippe; Sayas, Francisco-Javier
Theoretical and numerical study of a free boundary problem by boundary integral methods
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 35 (2001) no. 6 , p. 1137-1158
Zbl 0985.35114 | MR 1873520
URL stable : http://www.numdam.org/item?id=M2AN_2001__35_6_1137_0

Classification:  35R35,  41A15,  42A12,  45G05,  65R20
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.

Bibliographie

[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 0449.35105

[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 0804.65129

[3] M. Crouzeix, Variational approach of magnetic shaping problem. Eur. J. Mech. B/Fluids 10 (1991) 627-536. Zbl 0741.76089

[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 0741.76025

[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 679313 | Zbl 0564.49002

[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 0846.31005

[8] A. Henrot, Subsolutions and supersolutions in a free boundary problem. Ark. Mat. 32 (1994) 79-98. Zbl 0809.35172

[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 0672.65101

[10] R. Kress, Linear Integral Equations. Springer, New York (1989). MR 1007594 | Zbl 0671.45001

[11] W. Mclean and W.L. Wendland, Trigonometric approximation of solutions of periodic pseudodifferential equations. Oper. Theory: Adv. Appl. 41 (1989) 359-383. Zbl 0693.65093

[12] S. Mikhlin and S. Prößdorf, Singular Integral Operators. Springer-Verlag, Berlin (1986). MR 867687

[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. Zbl 0792.65096

[15] S. Prößdorf and B. Silbermann, Numerical Analysis for Integral and Related Operator Equations. Akademie-Verlag, Berlin (1991). MR 1193030 | Zbl 0763.65103

[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 0796.65124

[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 0682.45001