volume | PIER Journals

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.

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, New York, 2001.

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

8. Steinbach, O., Numerische Naherungsverfahren fur elliptische Randwertprobleme, Advances in Numerical Mathematics, BG Teubner, Stuttgart, 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, New York, 1964.

16. Van Bladel, J. G., Electromagnetic Fields, 2nd Ed., IEEE Press, Piscataway, 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, New York, 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, New York, 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, New York, 1984.

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

Professor Luc Knockaert

Dr. Daniel De Zutter