Theoretical and numerical study of a free boundary problem by boundary integral methods

Michel Crouzeix; Philippe Féat; Francisco-Javier Sayas

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

  • Volume: 35, Issue: 6, page 1137-1158
  • ISSN: 0764-583X

Abstract

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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.

How to cite

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Crouzeix, Michel, Féat, Philippe, and Sayas, Francisco-Javier. "Theoretical and numerical study of a free boundary problem by boundary integral methods." ESAIM: Mathematical Modelling and Numerical Analysis 35.6 (2010): 1137-1158. <http://eudml.org/doc/197491>.

@article{Crouzeix2010,
abstract = { 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. },
author = {Crouzeix, Michel, Féat, Philippe, Sayas, Francisco-Javier},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Free boundary; spline; trigonometric polynomial.; electromagnetic shaping; boundary integral methods; collocation methods with trigonometric polynomial and spline curves},
language = {eng},
month = {3},
number = {6},
pages = {1137-1158},
publisher = {EDP Sciences},
title = {Theoretical and numerical study of a free boundary problem by boundary integral methods},
url = {http://eudml.org/doc/197491},
volume = {35},
year = {2010},
}

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: Mathematical Modelling and Numerical Analysis
DA - 2010/3//
PB - EDP Sciences
VL - 35
IS - 6
SP - 1137
EP - 1158
AB - 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.
LA - eng
KW - Free boundary; spline; trigonometric polynomial.; electromagnetic shaping; boundary integral methods; collocation methods with trigonometric polynomial and spline curves
UR - http://eudml.org/doc/197491
ER -

References

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  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.  
  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.  
  3. M. Crouzeix, Variational approach of magnetic shaping problem. Eur. J. Mech. B/Fluids10 (1991) 627-536.  
  4. J. Descloux, Stability of solutions of the bidimensional magnetic shaping problem in absence of surface tension. Eur. J. Mech. B/Fluids10 (1991) 513-526.  
  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).  
  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.  
  8. A. Henrot, Subsolutions and supersolutions in a free boundary problem. Ark. Mat.32 (1994) 79-98.  
  9. A. Henrot and M. Pierre, Un problème inverse en formage des métaux liquides. RAIRO Modél. Math. Anal. Numér.23 (1989) 155-177.  
  10. R. Kress, Linear Integral Equations. Springer, New York (1989).  
  11. W. McLean and W.L. Wendland, Trigonometric approximation of solutions of periodic pseudodifferential equations. Oper. Theory: Adv. Appl.41 (1989) 359-383.  
  12. S. Mikhlin and S. Prößdorf, Singular Integral Operators. Springer-Verlag, Berlin (1986).  
  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.  
  15. S. Prößdorf and B. Silbermann, Numerical Analysis for Integral and Related Operator Equations. Akademie-Verlag, Berlin (1991).  
  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.  
  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.  

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