Vol. 7
Latest Volume
All Volumes
PIERB 109 [2024] PIERB 108 [2024] PIERB 107 [2024] PIERB 106 [2024] PIERB 105 [2024] PIERB 104 [2024] PIERB 103 [2023] PIERB 102 [2023] PIERB 101 [2023] PIERB 100 [2023] PIERB 99 [2023] PIERB 98 [2023] PIERB 97 [2022] PIERB 96 [2022] PIERB 95 [2022] PIERB 94 [2021] PIERB 93 [2021] PIERB 92 [2021] PIERB 91 [2021] PIERB 90 [2021] PIERB 89 [2020] PIERB 88 [2020] PIERB 87 [2020] PIERB 86 [2020] PIERB 85 [2019] PIERB 84 [2019] PIERB 83 [2019] PIERB 82 [2018] PIERB 81 [2018] PIERB 80 [2018] PIERB 79 [2017] PIERB 78 [2017] PIERB 77 [2017] PIERB 76 [2017] PIERB 75 [2017] PIERB 74 [2017] PIERB 73 [2017] PIERB 72 [2017] PIERB 71 [2016] PIERB 70 [2016] PIERB 69 [2016] PIERB 68 [2016] PIERB 67 [2016] PIERB 66 [2016] PIERB 65 [2016] PIERB 64 [2015] PIERB 63 [2015] PIERB 62 [2015] PIERB 61 [2014] PIERB 60 [2014] PIERB 59 [2014] PIERB 58 [2014] PIERB 57 [2014] PIERB 56 [2013] PIERB 55 [2013] PIERB 54 [2013] PIERB 53 [2013] PIERB 52 [2013] PIERB 51 [2013] PIERB 50 [2013] PIERB 49 [2013] PIERB 48 [2013] PIERB 47 [2013] PIERB 46 [2013] PIERB 45 [2012] PIERB 44 [2012] PIERB 43 [2012] PIERB 42 [2012] PIERB 41 [2012] PIERB 40 [2012] PIERB 39 [2012] PIERB 38 [2012] PIERB 37 [2012] PIERB 36 [2012] PIERB 35 [2011] PIERB 34 [2011] PIERB 33 [2011] PIERB 32 [2011] PIERB 31 [2011] PIERB 30 [2011] PIERB 29 [2011] PIERB 28 [2011] PIERB 27 [2011] PIERB 26 [2010] PIERB 25 [2010] PIERB 24 [2010] PIERB 23 [2010] PIERB 22 [2010] PIERB 21 [2010] PIERB 20 [2010] PIERB 19 [2010] PIERB 18 [2009] PIERB 17 [2009] PIERB 16 [2009] PIERB 15 [2009] PIERB 14 [2009] PIERB 13 [2009] PIERB 12 [2009] PIERB 11 [2009] PIERB 10 [2008] PIERB 9 [2008] PIERB 8 [2008] PIERB 7 [2008] PIERB 6 [2008] PIERB 5 [2008] PIERB 4 [2008] PIERB 3 [2008] PIERB 2 [2008] PIERB 1 [2008]
2008-04-08
On the Complex Symmetry of the Poincar'E-Steklov Operator
By
Progress In Electromagnetics Research B, Vol. 7, 145-157, 2008
Abstract
Employing Lorentz reciprocity and the Stratton-Chu formalism it is shown that the Poincare-Steklov or admittance operator can be interpreted as a complex symmetric operator mapping the tangential electric field (instead of the equivalent magnetic current) onto the equivalent electric current. We show that the pertinent block Calder´on projectors can be reformulated as operators with a block Hamiltonian structure. This leads to an explicitly complex symmetric Schur complement expression for both the interior and exterior admittance operators.
Citation
Luc Knockaert, and Daniel De Zutter, "On the Complex Symmetry of the Poincar'E-Steklov Operator," Progress In Electromagnetics Research B, Vol. 7, 145-157, 2008.
doi:10.2528/PIERB08022102
References

1. Stratton, J. A. and L. J. Chu, "Diffraction theory of electromagnetic waves," Phys. Rev., Vol. 56, No. 1, 99-107, 1939.
doi:10.1103/PhysRev.56.99

2. Hsiao, G. C. and R. E. Kleinman, "Mathematical foundations for error estimation in numerical solutions of integral equations in electromagnetics," IEEE Trans. Antennas Propagat., Vol. 45, No. 3, 316-328, 1997.
doi:10.1109/8.558648

3. Tai, C. T., "Direct integration of field equations," Progress In Electromagnetics Research, Vol. 28, 339-359, 2000.
doi:10.2528/PIER99101401

4. Borel, S., D. P. Levadoux, and F. Alouges, "A new wellconditioned integral formulation for Maxwell equations in three dimensions," IEEE Trans. Antennas Propagat., Vol. 53, No. 9, 2995-3004, 2005.
doi:10.1109/TAP.2005.854561

5. Sheng, X. Q., J. M. Jin, J. Song, W. C. Chew, and C. C. Lu, "Solution of combined-field integral equation using multilevel fast multipole algorithm for scattering by homogeneous bodies," IEEE Trans. Antennas Propagat., Vol. 46, No. 11, 1718-1726, 1998.
doi:10.1109/8.736628

6. Harrington, R. F., Time-harmonic Electromagnetic Fields, Wiley, 2001.

7. Sauter, S. and C. Schwab, Randelementmethoden, BG Teubner, 2004.

8. Steinbach, O., Numerische Naherungsverfahren fur elliptische Randwertprobleme, Advances in Numerical Mathematics, BG Teubner, 2003.

9. Hiptmair, R., "Coupling of finite elements and boundary elements in electromagnetic scattering," SIAM J. Numer. Anal., Vol. 41, 919-944, 2003.
doi:10.1137/S0036142901397757

10. Buffa, A. and R. Hiptmair, "Galerkin boundary element methods for electromagnetic scattering," Topics in Computational Wave Propagation. Direct and Inverse Problems, Vol. 31, 83-124, 2003.

11. de La Bourdonnaye, A., "Some formulations coupling finite element and integral equation methods for Helmholtz equation and electromagnetism," Numer. Math., Vol. 69, No. 3, 257-268, 1995.
doi:10.1007/s002110050091

12. Knockaert, L., D. De Zutter, G. Lippens, and H. Rogier, "On the Schur complement form of the Dirichlet-to-Neumann operator," Wave Motion, Vol. 45, No. 3, 309-324, 2008.
doi:10.1016/j.wavemoti.2007.07.004

13. Crabtree, D. E. and E. V. Haynsworth, "An identity for the Schur complement of a matrix," Proc. Amer. Math. Soc., Vol. 22, 364-366, 1969.
doi:10.2307/2037057

14. Corach, G., A. Maestripieri, and D. Stojanoff, "Generalized Schur complements and oblique projections," Lin. Alg. Appl., Vol. 341, 259-272, 2002.
doi:10.1016/S0024-3795(01)00384-6

15. Van Bladel, J., Electromagnetic Fields, McGraw-Hill, 1964.

16. Van Bladel, J. G., Electromagnetic Fields, 2nd Ed., IEEE Press, 2007.

17. Garcia, S. R. and M. Putinar, "Complex symmetric operators and applications," Trans. Amer. Math. Soc., Vol. 358, No. 3, 1285-1315, 2005.
doi:10.1090/S0002-9947-05-03742-6

18. Meyer, K. R. and G. R. Hall, Introduction to Hamiltonian Dynamical Systems and the N-body Problem. Applied Mathematical Sciences, Vol. 90, Springer, 1992.

19. Lin, W. W. and C. S. Wang, "On computing stable Lagrangian subspaces of Hamiltonian matrices and symplectic pencils," SIAM J. Matrix Anal. Appl., Vol. 18, 590-614, 1997.
doi:10.1137/S0895479894272712

20. Laub, A., "A Schur method for solving algebraic Riccati equations," IEEE Trans. Automat. Control, Vol. 24, 913-921, 1979.
doi:10.1109/TAC.1979.1102178

21. Hanson, G. W. and A. B. Yakovlev, Operator Theory for Electromagnetics. An Introduction, Springer, 2002.

22. Deschamps, G. A., "Electromagnetics and differential forms," Proc. IEEE, Vol. 69, No. 6, 676-696, 1981.
doi:10.1109/PROC.1981.12048

23. Colton, D. and R. Kress, Integral Equation Methods in Scattering Theory, Wiley, 1984.

24. Jones, D. S., Acoustic and Electromagnetic Waves, Clarendon Press, 1986.