Journal of Integral Equations and Applications

A well-conditioned boundary integral equation for transmission problems of electromagnetism

David Levadoux, Florence Millot, and Sébastien Pernet

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Abstract

We propose a new well-conditioned boundary integral equation to solve transmission problems of electromagnetism. This equation is well posed and appears as a compact perturbation of the identity leading to fast iterative solutions without the help of any preconditioner. Some numerical experiments confirm this result.

Article information

Source
J. Integral Equations Applications, Volume 27, Number 3 (2015), 431-454.

Dates
First available in Project Euclid: 17 December 2015

Permanent link to this document
https://projecteuclid.org/euclid.jiea/1450388943

Digital Object Identifier
doi:10.1216/JIE-2015-27-3-431

Mathematical Reviews number (MathSciNet)
MR3435808

Zentralblatt MATH identifier
1332.31005

Subjects
Primary: 31B10: Integral representations, integral operators, integral equations methods 65F08: Preconditioners for iterative methods 76M15: Boundary element methods 78M16: Multipole methods

Keywords
Integral equation boundary element method transmission problem preconditioner GMRES fast multipole method

Citation

Levadoux, David; Millot, Florence; Pernet, Sébastien. A well-conditioned boundary integral equation for transmission problems of electromagnetism. J. Integral Equations Applications 27 (2015), no. 3, 431--454. doi:10.1216/JIE-2015-27-3-431. https://projecteuclid.org/euclid.jiea/1450388943


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References

  • F.P. Andriulli, Well-posed boundary element formulations in electromagnetics, Ph.D. dissertation, University of Michigan.
  • F. Andriulli, K. Cools, H. Bagci, F. Olyslager, A. Buffa, S. Christiansen and E. Michielssen, A multiplicative Calderon preconditioner for the electric field integral equation, IEEE Trans. Ant. Prop. 56 (2008), 2398–2412.
  • S. Borel, D. Levadoux and F. Alouges, A new well-conditioned integral formulation for Maxwell equations in three-dimensions, IEEE Trans. Ant. Prop. 53 (2005), 2995–3004.
  • Y. Boubendir, O. Bruno, D. Levadoux and C. Turc, Integral equations requiring small numbers of Krylov-subspace iterations for two-dimensional smooth penetrable scattering problems, Appl. Numer. Math. 95 (2015), 82–98.
  • ––––, Regularized combined field integral equations for acoustic transmission problems, SIAM J. Appl. Math 75 (2015), 929–952.
  • O.P. Bruno, Computational electromagnetics and acoustics: Highorder solvers, highfrequency configurations, high-order surface representations, Oberwolfach Reports 5 (2007), 284–287.
  • O. Bruno, T. Elling, R. Paffenroth and C. Turc, Electromagnetic integral equations requiring small numbers of Krylov-subspace iterations, J. Comp. Phys. 228 (2009), 6169–6183.
  • A. Buffa, R. Hiptmair, T. von Petersdorff and C. Schwab, Boundary element methods for Maxwell transmission problems in Lipschitz domains, Numer. Math. 95 (2003), 459–485.
  • S.L. Campbell, I.C.F. Ipsen, C.T. Kelley and C.D. Meyer, GMRES and the minimal polynomial, BIT Numer. Math. 36, Springer, Netherlands, 1996.
  • S.L. Campbell, I.C.F. Ipsen, C.T. Kelley, C.D. Meyer and Z.Q. Xue, Convergence estimates for solution of integral equations with GMRES, J. Integral Equations Applications 8 (1996), 19–34.
  • B. Carpentieri, I.S. Duff and L. Giraud, Sparse pattern selection strategies for robust frobenius-norm minimization preconditioners in electromagnetism, Numer. Lin. Alg. Appl. 7 (2000), 667–685.
  • D. Colton and P. Kress, Integral equation methods in scattering, Wiley & Sons, New York, 1983.
  • ––––, Inverse acoustic and electromagnetic scattering theory, Springer-Verlag, New York, 1992.
  • H. Contopanagos, D. Dembart, M. Epton, J. Ottusch, V. Rokhlin, J. Visher and S. Vandzura, Well-conditioned boundary integral equations for three-dimensional electromagnetic scattering, IEEE Trans. Ant. Prop. 50 (2002), 1824–1830.
  • M. Darbas, Generalized CFIE for the iterative solution of 3-D Maxwell equations, Appl. Math. Lett. 19 (2006), 834–839.
  • B. Jung, T. Sarkar and Y. Chung, A survey of various frequency domain integral equations for the analysis of scattering from three-dimensional dielectric objects, PIER 36 (2002), 193–246.
  • D. Levadoux, Etude d'une équation intégrale adaptée à la résolution haute fréquence de l'équation de helmholtz, Ph.D. thesis, Paris, 2001.
  • ––––, A new integral formalism for transmission problems of electromagnetism, 8th Inter. Conf. Math. Numer. Aspects Waves, Reading, UK, July 23-27, 2007, 90–92.
  • ––––, Some preconditioners for the CFIE equation of electromagnetism, Math. Meth. Appl. Sci. 31 (2008), 2015–2028.
  • D. Levadoux and B.L. Michielsen, Analysis of a boundary integral equation for high frequency helmholtz problems, in Proc. 4th Inter. Conf. Math. Numer. Aspects Wave Propag., Golden, Colorado, 1998, 765–767.
  • ––––, Nouvelles formuations intégrales pour les problèmes de diffraction d'ondes, Math. Model. Numer. Anal. 38 (2004), 157–175.
  • D. Levadoux, F. Millot and S. Pernet, New trends in the preconditioning of integral equations of electromagnetism, in Mathematics in industry–scientific computing in electical engineering, J. Roos and Luis R.J. Costa, ed., Springer, New York, 2010.
  • ––––, An unpreconditioned boundaryintegral for iterative solution of scattering problems with non-constant Leontovitch impedance boundary conditions, Comm. Comp. Phys. 15 (2014), 1431–1460.
  • J.C. Nédélec, Acoustic and electromagnetic equations: Integral representation for harmonic problems, Springer-Verlag, New York, 2001.
  • S. Pernet, A well-conditioned integral equation for iterative solution of scattering problems with a variable Leontovitch boundary condition, Math. Model. Numer. Anal. 44 (2010), 781–801.
  • E.P. Stephan, Boundary Integral equation for mixed boundary value problems, screen and transmission problems in $R^3$, Habilitationsschrift, Tech. Hochschule, Darmstadt, 1984.