Crouzeix, Michel; Féat, Philippe; Sayas, Francisco-Javier Theoretical and numerical study of a free boundary problem by boundary integral methods. (English) Zbl 0985.35114 M2AN, Math. Model. Numer. Anal. 35, No. 6, 1137-1158 (2001). Summary: 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 an 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. Cited in 7 Documents MSC: 35R35 Free boundary problems for PDEs 31A10 Integral representations, integral operators, integral equations methods in two dimensions 35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation 42A10 Trigonometric approximation 45G05 Singular nonlinear integral equations 65R20 Numerical methods for integral equations 41A15 Spline approximation Keywords:electromagnetic shaping; boundary integral methods; collocation methods with trigonometric polynomial and spline curves PDF BibTeX XML Cite \textit{M. Crouzeix} et al., M2AN, Math. Model. Numer. Anal. 35, No. 6, 1137--1158 (2001; Zbl 0985.35114) Full Text: DOI Numdam EuDML References: [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 · Zbl 0449.35105 · crelle:GDZPPN002198525 · eudml:152360 [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 · Zbl 0804.65129 · doi:10.1137/0731058 [3] M. Crouzeix , Variational approach of magnetic shaping problem . Eur. J. Mech. B/Fluids 10 ( 1991 ) 627 - 536 . Zbl 0741.76089 · 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 · 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 · 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 · Zbl 0846.31005 · crelle:GDZPPN002213540 · eudml:153808 [8] A. Henrot , Subsolutions and supersolutions in a free boundary problem . Ark. Mat. 32 ( 1994 ) 79 - 98 . Zbl 0809.35172 · Zbl 0809.35172 · doi:10.1007/BF02559524 [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 · Zbl 0672.65101 · eudml:193550 [10] R. Kress , Linear Integral Equations . Springer, New York ( 1989 ). MR 1007594 | Zbl 0671.45001 · 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 · 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 . Article | Zbl 0792.65096 · Zbl 0792.65096 · doi:10.1007/BF01385748 · eudml:133731 [15] S. Prößdorf and B. Silbermann , Numerical Analysis for Integral and Related Operator Equations . Akademie-Verlag, Berlin ( 1991 ). MR 1193030 | Zbl 0763.65103 · 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 · Zbl 0796.65124 · doi:10.1137/0730090 [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 . Article | Zbl 0682.45001 · Zbl 0682.45001 · doi:10.1216/JIE-1988-1-4-549 · minidml.mathdoc.fr This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.