June  2015, 8(3): 619-647. doi: 10.3934/dcdss.2015.8.619

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

Received  November 2013 Revised  June 2014 Published  October 2014

Various asymptotic models for thin conducting sheets in computational electromagnetics describe them as closed hyper-surfaces equipped with linear local transmission conditions for the traces of electric and magnetic fields. The transmission conditions turn out to be singularly perturbed with respect to limit values of parameters depending on sheet thickness and conductivity. We consider the reformulation of the resulting transmission problems into boundary integral equations (BIE) and their Galerkin discretization by means of low-order boundary elements. We establish stability of the BIE and provide a priori $h$-convergence estimates.
Citation: Kersten Schmidt, Ralf Hiptmair. Asymptotic boundary element methods for thin conducting sheets. Discrete & Continuous Dynamical Systems - S, 2015, 8 (3) : 619-647. doi: 10.3934/dcdss.2015.8.619
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show all references

References:
[1]

Academic Press, New-York, London, 1975.  Google Scholar

[2]

in Partial differential equations: Theory and numerical solution., 406 of Chapman & Hall/CRC Res. Notes Math., 2000, 10-24.  Google Scholar

[3]

Asymptotic Analysis, 57 (2008), 199-227.  Google Scholar

[4]

SIAM J. Appl. Math., 56 (1996), 1664-1693. doi: 10.1137/S0036139995281822.  Google Scholar

[5]

3rd edition, Cambridge University Press, 2007. doi: 10.1088/0957-0233/13/9/704.  Google Scholar

[6]

URL http://www.concepts.math.ethz.ch, 2014. Google Scholar

[7]

Technical report, Ecole Polytechnique Paris, 1993, Rapport interne du C.M.A.P. Google Scholar

[8]

Math. Model. Numer. Anal., 36 (2002), 937-951. doi: 10.1051/m2an:2002036.  Google Scholar

[9]

Math. Models Methods Appl. Sci, 15 (2005), 1273-1300. doi: 10.1142/S021820250500073X.  Google Scholar

[10]

Math. Models Meth. Appl. Sci., 18 (2008), 1787-1827. doi: 10.1142/S0218202508003194.  Google Scholar

[11]

Rend. Lincei (5), 11 (1902), 75-81. Google Scholar

[12]

Magnetics, IEEE Transactions on, 31 (1995), 1319-1324. doi: 10.1109/20.376271.  Google Scholar

[13]

Cambridge University Press, 2000.  Google Scholar

[14]

Magnetics, IEEE Transactions on, 26 (1990), 2379-2381. doi: 10.1109/20.104737.  Google Scholar

[15]

Math. Meth. Appl. Sci., 31 (2008), 443-479. doi: 10.1002/mma.923.  Google Scholar

[16]

Math. Meth. Appl. Sci., 32 (2009), 435-453. doi: 10.1002/mma.1045.  Google Scholar

[17]

Springer-Verlag, Heidelberg, 2011. doi: 10.1007/978-3-540-68093-2.  Google Scholar

[18]

SIAM J. Appl. Math, 73 (2013), 1980-2003. doi: 10.1137/120901398.  Google Scholar

[19]

Preprint 2013-15, Inst. f. Mathematik, TU Berlin, 2013. Google Scholar

[20]

INS Report 1102, Institut for Numerical Simulation, University of Bonn, 2011. Google Scholar

[21]

Z. Angew. Math. Phys., 61 (2010), 603-626. doi: 10.1007/s00033-009-0043-x.  Google Scholar

[22]

ESAIM: M2AN, 45 (2011), 1115-1140. doi: 10.1051/m2an/2011009.  Google Scholar

[23]

B.G. Teubner-Verlag, 2003. Google Scholar

[24]

Technika, Kiev, 1974, (in Russian). Google Scholar

[25]

Math. Meth. Appl. Sci., 22 (1999), 587-603. Google Scholar

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