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On Maxwell's and Poincaré's constants
Asymptotic boundary element methods for thin conducting sheets
1. | Research Center MATHEON, Institut für Mathematik, Technische Universität Berlin, 10623 Berlin, Germany |
2. | Seminar for Applied Mathematics, ETH Zurich, 8092 Zürich, Switzerland |
References:
[1] |
R. A. Adams, Sobolev Spaces, Academic Press, New-York, London, 1975. |
[2] |
A. Bendali, Boundary element solution of scattering problems relative to a generalized impedance boundary condition, in Partial differential equations: Theory and numerical solution., 406 of Chapman & Hall/CRC Res. Notes Math., 2000, 10-24. |
[3] |
A. Bendali and K. Lemrabet, Asymptotic analysis of the scattering of a time-harmonic electromagnetic wave by a perfectly conducting metal coated with a thin dielectric shell, Asymptotic Analysis, 57 (2008), 199-227. |
[4] |
A. Bendali and K. Lemrabet, The effect of a thin coating on the scattering of a time-harmonic wave for the Helmholtz equation, SIAM J. Appl. Math., 56 (1996), 1664-1693.
doi: 10.1137/S0036139995281822. |
[5] |
D. Braess, Finite Elements: Theory, Fast Solvers, and Applications in Solid Mechanics, 3rd edition, Cambridge University Press, 2007.
doi: 10.1088/0957-0233/13/9/704. |
[6] |
Concepts Development Team, Webpage of Numerical C++ Library Concepts 2, URL http://www.concepts.math.ethz.ch, 2014. |
[7] |
B. Engquist and J.-C. Nédélec, Effective boundary conditions for acoustic and electromagnetic scattering in thin layers, Technical report, Ecole Polytechnique Paris, 1993, Rapport interne du C.M.A.P. |
[8] |
P. Frauenfelder and C. Lage, Concepts - An Object-Oriented Software Package for Partial Differential Equations, Math. Model. Numer. Anal., 36 (2002), 937-951.
doi: 10.1051/m2an:2002036. |
[9] |
H. Haddar, P. Joly and H. Nguyen, Generalized impedance boundary conditions for scattering by strongly absorbing obstacles: the scalar case, Math. Models Methods Appl. Sci, 15 (2005), 1273-1300.
doi: 10.1142/S021820250500073X. |
[10] |
H. Haddar, P. Joly and H.-M. Nguyen, Generalized impedance boundary conditions for scattering problems from strongly absorbing obstacles: the case of Maxwell's equations, Math. Models Meth. Appl. Sci., 18 (2008), 1787-1827.
doi: 10.1142/S0218202508003194. |
[11] |
T. Levi-Civita, La teoria elettrodinamica di Hertz di fronte ai fenomeni di induzione, Rend. Lincei (5), 11 (1902), 75-81. |
[12] |
I. Mayergoyz and G. Bedrosian, On calculation of 3-D eddy currents in conducting and magnetic shells, Magnetics, IEEE Transactions on, 31 (1995), 1319-1324.
doi: 10.1109/20.376271. |
[13] |
W. McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, 2000. |
[14] |
T. Nakata, N. Takahashi, K. Fujiwara and Y. Shiraki, 3D magnetic field analysis using special elements, Magnetics, IEEE Transactions on, 26 (1990), 2379-2381.
doi: 10.1109/20.104737. |
[15] |
C. Poignard, Asymptotics for steady-state voltage potentials in a bidimensional highly contrasted medium with thin layer, Math. Meth. Appl. Sci., 31 (2008), 443-479.
doi: 10.1002/mma.923. |
[16] |
C. Poignard, Approximate transmission conditions through a weakly oscillating thin layer, Math. Meth. Appl. Sci., 32 (2009), 435-453.
doi: 10.1002/mma.1045. |
[17] |
S. Sauter and C. Schwab, Boundary Element Methods, Springer-Verlag, Heidelberg, 2011.
doi: 10.1007/978-3-540-68093-2. |
[18] |
K. Schmidt and A. Chernov, A unified analysis of transmission conditions for thin conducting sheets in the time-harmonic eddy current model, SIAM J. Appl. Math, 73 (2013), 1980-2003.
doi: 10.1137/120901398. |
[19] |
K. Schmidt and R. Hiptmair, Asymptotic boundary element methods for thin conducting sheets, Preprint 2013-15, Inst. f. Mathematik, TU Berlin, 2013. |
[20] |
K. Schmidt and A. Chernov, Robust Families of Transmission Conditions of High Order for Thin Conducting Sheets, INS Report 1102, Institut for Numerical Simulation, University of Bonn, 2011. |
[21] |
K. Schmidt and S. Tordeux, Asymptotic modelling of conductive thin sheets, Z. Angew. Math. Phys., 61 (2010), 603-626.
doi: 10.1007/s00033-009-0043-x. |
[22] |
K. Schmidt and S. Tordeux, High order transmission conditions for thin conductive sheets in magneto-quasistatics, ESAIM: M2AN, 45 (2011), 1115-1140.
doi: 10.1051/m2an/2011009. |
[23] |
O. Steinbach, Numerische Näherungsverfahren für elliptische Randwertprobleme. Finite Elemente und Randelemente, B.G. Teubner-Verlag, 2003. |
[24] |
O. Tozoni and I. Mayergoyz, Calculation of three-dimensional electromagnetic fields, Technika, Kiev, 1974, (in Russian). |
[25] |
L. Vernhet, generalized impedance boundary condition, Math. Meth. Appl. Sci., 22 (1999), 587-603. |
show all references
References:
[1] |
R. A. Adams, Sobolev Spaces, Academic Press, New-York, London, 1975. |
[2] |
A. Bendali, Boundary element solution of scattering problems relative to a generalized impedance boundary condition, in Partial differential equations: Theory and numerical solution., 406 of Chapman & Hall/CRC Res. Notes Math., 2000, 10-24. |
[3] |
A. Bendali and K. Lemrabet, Asymptotic analysis of the scattering of a time-harmonic electromagnetic wave by a perfectly conducting metal coated with a thin dielectric shell, Asymptotic Analysis, 57 (2008), 199-227. |
[4] |
A. Bendali and K. Lemrabet, The effect of a thin coating on the scattering of a time-harmonic wave for the Helmholtz equation, SIAM J. Appl. Math., 56 (1996), 1664-1693.
doi: 10.1137/S0036139995281822. |
[5] |
D. Braess, Finite Elements: Theory, Fast Solvers, and Applications in Solid Mechanics, 3rd edition, Cambridge University Press, 2007.
doi: 10.1088/0957-0233/13/9/704. |
[6] |
Concepts Development Team, Webpage of Numerical C++ Library Concepts 2, URL http://www.concepts.math.ethz.ch, 2014. |
[7] |
B. Engquist and J.-C. Nédélec, Effective boundary conditions for acoustic and electromagnetic scattering in thin layers, Technical report, Ecole Polytechnique Paris, 1993, Rapport interne du C.M.A.P. |
[8] |
P. Frauenfelder and C. Lage, Concepts - An Object-Oriented Software Package for Partial Differential Equations, Math. Model. Numer. Anal., 36 (2002), 937-951.
doi: 10.1051/m2an:2002036. |
[9] |
H. Haddar, P. Joly and H. Nguyen, Generalized impedance boundary conditions for scattering by strongly absorbing obstacles: the scalar case, Math. Models Methods Appl. Sci, 15 (2005), 1273-1300.
doi: 10.1142/S021820250500073X. |
[10] |
H. Haddar, P. Joly and H.-M. Nguyen, Generalized impedance boundary conditions for scattering problems from strongly absorbing obstacles: the case of Maxwell's equations, Math. Models Meth. Appl. Sci., 18 (2008), 1787-1827.
doi: 10.1142/S0218202508003194. |
[11] |
T. Levi-Civita, La teoria elettrodinamica di Hertz di fronte ai fenomeni di induzione, Rend. Lincei (5), 11 (1902), 75-81. |
[12] |
I. Mayergoyz and G. Bedrosian, On calculation of 3-D eddy currents in conducting and magnetic shells, Magnetics, IEEE Transactions on, 31 (1995), 1319-1324.
doi: 10.1109/20.376271. |
[13] |
W. McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, 2000. |
[14] |
T. Nakata, N. Takahashi, K. Fujiwara and Y. Shiraki, 3D magnetic field analysis using special elements, Magnetics, IEEE Transactions on, 26 (1990), 2379-2381.
doi: 10.1109/20.104737. |
[15] |
C. Poignard, Asymptotics for steady-state voltage potentials in a bidimensional highly contrasted medium with thin layer, Math. Meth. Appl. Sci., 31 (2008), 443-479.
doi: 10.1002/mma.923. |
[16] |
C. Poignard, Approximate transmission conditions through a weakly oscillating thin layer, Math. Meth. Appl. Sci., 32 (2009), 435-453.
doi: 10.1002/mma.1045. |
[17] |
S. Sauter and C. Schwab, Boundary Element Methods, Springer-Verlag, Heidelberg, 2011.
doi: 10.1007/978-3-540-68093-2. |
[18] |
K. Schmidt and A. Chernov, A unified analysis of transmission conditions for thin conducting sheets in the time-harmonic eddy current model, SIAM J. Appl. Math, 73 (2013), 1980-2003.
doi: 10.1137/120901398. |
[19] |
K. Schmidt and R. Hiptmair, Asymptotic boundary element methods for thin conducting sheets, Preprint 2013-15, Inst. f. Mathematik, TU Berlin, 2013. |
[20] |
K. Schmidt and A. Chernov, Robust Families of Transmission Conditions of High Order for Thin Conducting Sheets, INS Report 1102, Institut for Numerical Simulation, University of Bonn, 2011. |
[21] |
K. Schmidt and S. Tordeux, Asymptotic modelling of conductive thin sheets, Z. Angew. Math. Phys., 61 (2010), 603-626.
doi: 10.1007/s00033-009-0043-x. |
[22] |
K. Schmidt and S. Tordeux, High order transmission conditions for thin conductive sheets in magneto-quasistatics, ESAIM: M2AN, 45 (2011), 1115-1140.
doi: 10.1051/m2an/2011009. |
[23] |
O. Steinbach, Numerische Näherungsverfahren für elliptische Randwertprobleme. Finite Elemente und Randelemente, B.G. Teubner-Verlag, 2003. |
[24] |
O. Tozoni and I. Mayergoyz, Calculation of three-dimensional electromagnetic fields, Technika, Kiev, 1974, (in Russian). |
[25] |
L. Vernhet, generalized impedance boundary condition, Math. Meth. Appl. Sci., 22 (1999), 587-603. |
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