Abstract

We introduce a simple method to accelerate the convergence of iterative solvers of the frequency-domain Maxwell’s equations for deep-subwavelength structures. Using the continuity equation, the method eliminates the high multiplicity of near-zero eigenvalues of the operator while leaving the operator nearly positive-definite. The impact of the modified eigenvalue distribution on the accelerated convergence is explained by visualizing residual vectors and residual polynomials.

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  1. N. J. Champagne, J. G. Berryman, and H. M. Buettner, “FDFD: A 3D finite-difference frequency-domain code for electromagnetic induction tomography,” J. Comput. Phys.170, 830–848 (2001).
    [CrossRef]
  2. G. Veronis and S. Fan, “Overview of simulation techniques for plasmonic devices,” in “Surface Plasmon Nanophotonics,” M. Brongersma and P. Kik, eds. (Springer, 2007), pp. 169–182.
    [CrossRef]
  3. U. S. Inan and R. A. Marshall, Numerical Electromagnetics: The FDTD Method (Cambridge University, 2011). Ch. 14.
    [CrossRef]
  4. J.-M. Jin, The Finite Element Method in Electromagnetics, 2nd ed. (Wiley, 2002).
  5. V. Simoncini and D. B. Szyld, “Recent computational developments in Krylov subspace methods for linear systems,” Numer. Linear Algebra Appl.14, 1–59 (2007).
    [CrossRef]
  6. W. Cai, W. Shin, S. Fan, and M. L. Brongersma, “Elements for plasmonic nanocircuits with three-dimensional slot waveguides,” Adv. Mater.22, 5120–5124 (2010).
    [CrossRef] [PubMed]
  7. G. Veronis and S. Fan, “Bends and splitters in metal-dielectric-metal subwavelength plasmonic waveguides,” Appl. Phys. Lett.87, 131102–3 (2005).
    [CrossRef]
  8. L. Verslegers, P. Catrysse, Z. Yu, W. Shin, Z. Ruan, and S. Fan, “Phase front design with metallic pillar arrays,” Opt. Lett.35, 844–846 (2010).
    [CrossRef] [PubMed]
  9. J. T. Smith, “Conservative modeling of 3-D electromagnetic fields, Part II: Biconjugate gradient solution and an accelerator,” Geophys.61, 1319–1324 (1996).
    [CrossRef]
  10. G. A. Newman and D. L. Alumbaugh, “Three-dimensional induction logging problems, Part 2: A finite-difference solution,” Geophys.67, 484–491 (2002).
    [CrossRef]
  11. C. J. Weiss and G. A. Newman, “Electromagnetic induction in a generalized 3D anisotropic earth, Part 2: The LIN preconditioner,” Geophys.68, 922–930 (2003).
    [CrossRef]
  12. V. L. Druskin, L. A. Knizhnerman, and P. Lee, “New spectral Lanczos decomposition method for induction modeling in arbitrary 3-D geometry,” Geophys.64, 701–706 (1999).
    [CrossRef]
  13. E. Haber, U. M. Ascher, D. A. Aruliah, and D. W. Oldenburg, “Fast simulation of 3D electromagnetic problems using potentials,” J. Comput. Phys.163, 150–171 (2000).
    [CrossRef]
  14. J. Hou, R. Mallan, and C. Torres-Verdin, “Finite-difference simulation of borehole EM measurements in 3D anisotropic media using coupled scalar-vector potentials,” Geophys.71, G225–G233 (2006).
    [CrossRef]
  15. R. Hiptmair, F. Kramer, and J. Ostrowski, “A robust Maxwell formulation for all frequencies,” IEEE Trans. Magn.44, 682–685 (2008).
    [CrossRef]
  16. K. Beilenhoff, W. Heinrich, and H. Hartnagel, “Improved finite-difference formulation in frequency domain for three-dimensional scattering problems,” IEEE Trans. Microwave Theory Tech.40, 540–546 (1992).
    [CrossRef]
  17. A. Christ and H. L. Hartnagel, “Three-dimensional finite-difference method for the analysis of microwave-device embedding,” IEEE Trans. Microwave Theory Tech.35, 688–696 (1987).
    [CrossRef]
  18. At the final stage of our work, we were made aware of a related work by M. Kordy, E. Cherkaev, and P. Wannamaker, “Schelkunoff potential for electromagnetic field: proof of existence and uniqueness” (to be published), where an equation similar to our Eq. (7) with s= −1 was developed.
  19. A. Jennings, “Influence of the eigenvalue spectrum on the convergence rate of the conjugate gradient method,” IMA J. Appl. Math.20, 61–72 (1977).
    [CrossRef]
  20. A. van der Sluis and H. van der Vorst, “The rate of convergence of conjugate gradients,” Numer. Math.48, 543–560 (1986).
    [CrossRef]
  21. S. L. Campbell, I. C. F. Ipsen, C. T. Kelley, and C. D. Meyer, “GMRES and the minimal polynomial,” BIT Numer. Math.36, 664–675 (1996).
    [CrossRef]
  22. S. Goossens and D. Roose, “Ritz and harmonic Ritz values and the convergence of FOM and GMRES,” Numer. Linear Algebra Appl.6, 281–293 (1999).
    [CrossRef]
  23. M. Benzi, G. H. Golub, and J. Liesen, “Numerical solution of saddle point problems,” Acta Numer.14, 1–137 (2005). Sec. 9.2.
    [CrossRef]
  24. B. Beckermann and A. B. J. Kuijlaars, “Superlinear convergence of conjugate gradients,” SIAM J. Numer. Anal.39, 300–329 (2001).
    [CrossRef]
  25. B. Beckermann and A. B. J. Kuijlaars, “On the sharpness of an asymptotic error estimate for conjugate gradients,” BIT Numer. Math.41, 856–867 (2001).
    [CrossRef]
  26. O. Axelsson, “Iteration number for the conjugate gradient method,” Math. Comput. Simulat.61, 421–435 (2003).
    [CrossRef]
  27. J. P. Webb, “The finite-element method for finding modes of dielectric-loaded cavities,” IEEE Trans. Microwave Theory Tech.33, 635–639 (1985).
    [CrossRef]
  28. F. Kikuchi, “Mixed and penalty formulations for finite element analysis of an eigenvalue problem in electromagnetism,” Comput. Method Appl. Mech. Eng.64, 509–521 (1987).
    [CrossRef]
  29. D. A. White and J. M. Koning, “Computing solenoidal eigenmodes of the vector Helmholtz equation: a novel approach,” IEEE Trans. Magn.38, 3420–3425 (2002).
    [CrossRef]
  30. N. W. Ashcroft and N. D. Mermin, Solid State Physics, 1st ed. (SaundersCollege, 1976), Ch. 8.
  31. To obtain 8/Δmin2, take the first equation of Eq. (4.29) in [32] and then multiply the extra factor μ= μ0to account for the difference between Eq. (4.17a) in [32] and Eq. (1) in the present paper.
  32. W. Shin and S. Fan, “Choice of the perfectly matched layer boundary condition for frequency-domain Maxwell’s equations solvers,” J. Comput. Phys.231, 3406–3431 (2012).
    [CrossRef]
  33. Y. Saad and M. H. Schultz, “GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems,” SIAM J. Sci. Stat. Comp.7, 856–869 (1986).
    [CrossRef]
  34. P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B6, 4370–4379 (1972).
    [CrossRef]
  35. E. D. Palik, ed., Handbook of Optical Constants of Solids (Academic Press, 1985).
  36. D. R. Lide, ed., CRC Handbook of Chemistry and Physics, 88th ed. (CRC, 2007).
  37. R. Freund and N. Nachtigal, “QMR: a quasi-minimal residual method for non-hermitian linear systems,” Numer. Math.60, 315–339 (1991).
    [CrossRef]
  38. R. W. Freund, G. H. Golub, and N. M. Nachtigal, “Iterative solution of linear systems,” Acta Numer.1, 57–100 (1992). Secs. 2.4 and 3.3.
    [CrossRef]
  39. J. W. Goodman, Introduction to Fourier Optics(Roberts & Company Publishers, 2005), 3rd ed. Sec. 2.3.2.

2012 (1)

W. Shin and S. Fan, “Choice of the perfectly matched layer boundary condition for frequency-domain Maxwell’s equations solvers,” J. Comput. Phys.231, 3406–3431 (2012).
[CrossRef]

2010 (2)

W. Cai, W. Shin, S. Fan, and M. L. Brongersma, “Elements for plasmonic nanocircuits with three-dimensional slot waveguides,” Adv. Mater.22, 5120–5124 (2010).
[CrossRef] [PubMed]

L. Verslegers, P. Catrysse, Z. Yu, W. Shin, Z. Ruan, and S. Fan, “Phase front design with metallic pillar arrays,” Opt. Lett.35, 844–846 (2010).
[CrossRef] [PubMed]

2008 (1)

R. Hiptmair, F. Kramer, and J. Ostrowski, “A robust Maxwell formulation for all frequencies,” IEEE Trans. Magn.44, 682–685 (2008).
[CrossRef]

2007 (1)

V. Simoncini and D. B. Szyld, “Recent computational developments in Krylov subspace methods for linear systems,” Numer. Linear Algebra Appl.14, 1–59 (2007).
[CrossRef]

2006 (1)

J. Hou, R. Mallan, and C. Torres-Verdin, “Finite-difference simulation of borehole EM measurements in 3D anisotropic media using coupled scalar-vector potentials,” Geophys.71, G225–G233 (2006).
[CrossRef]

2005 (2)

G. Veronis and S. Fan, “Bends and splitters in metal-dielectric-metal subwavelength plasmonic waveguides,” Appl. Phys. Lett.87, 131102–3 (2005).
[CrossRef]

M. Benzi, G. H. Golub, and J. Liesen, “Numerical solution of saddle point problems,” Acta Numer.14, 1–137 (2005). Sec. 9.2.
[CrossRef]

2003 (2)

O. Axelsson, “Iteration number for the conjugate gradient method,” Math. Comput. Simulat.61, 421–435 (2003).
[CrossRef]

C. J. Weiss and G. A. Newman, “Electromagnetic induction in a generalized 3D anisotropic earth, Part 2: The LIN preconditioner,” Geophys.68, 922–930 (2003).
[CrossRef]

2002 (2)

G. A. Newman and D. L. Alumbaugh, “Three-dimensional induction logging problems, Part 2: A finite-difference solution,” Geophys.67, 484–491 (2002).
[CrossRef]

D. A. White and J. M. Koning, “Computing solenoidal eigenmodes of the vector Helmholtz equation: a novel approach,” IEEE Trans. Magn.38, 3420–3425 (2002).
[CrossRef]

2001 (3)

B. Beckermann and A. B. J. Kuijlaars, “Superlinear convergence of conjugate gradients,” SIAM J. Numer. Anal.39, 300–329 (2001).
[CrossRef]

B. Beckermann and A. B. J. Kuijlaars, “On the sharpness of an asymptotic error estimate for conjugate gradients,” BIT Numer. Math.41, 856–867 (2001).
[CrossRef]

N. J. Champagne, J. G. Berryman, and H. M. Buettner, “FDFD: A 3D finite-difference frequency-domain code for electromagnetic induction tomography,” J. Comput. Phys.170, 830–848 (2001).
[CrossRef]

2000 (1)

E. Haber, U. M. Ascher, D. A. Aruliah, and D. W. Oldenburg, “Fast simulation of 3D electromagnetic problems using potentials,” J. Comput. Phys.163, 150–171 (2000).
[CrossRef]

1999 (2)

V. L. Druskin, L. A. Knizhnerman, and P. Lee, “New spectral Lanczos decomposition method for induction modeling in arbitrary 3-D geometry,” Geophys.64, 701–706 (1999).
[CrossRef]

S. Goossens and D. Roose, “Ritz and harmonic Ritz values and the convergence of FOM and GMRES,” Numer. Linear Algebra Appl.6, 281–293 (1999).
[CrossRef]

1996 (2)

S. L. Campbell, I. C. F. Ipsen, C. T. Kelley, and C. D. Meyer, “GMRES and the minimal polynomial,” BIT Numer. Math.36, 664–675 (1996).
[CrossRef]

J. T. Smith, “Conservative modeling of 3-D electromagnetic fields, Part II: Biconjugate gradient solution and an accelerator,” Geophys.61, 1319–1324 (1996).
[CrossRef]

1992 (2)

K. Beilenhoff, W. Heinrich, and H. Hartnagel, “Improved finite-difference formulation in frequency domain for three-dimensional scattering problems,” IEEE Trans. Microwave Theory Tech.40, 540–546 (1992).
[CrossRef]

R. W. Freund, G. H. Golub, and N. M. Nachtigal, “Iterative solution of linear systems,” Acta Numer.1, 57–100 (1992). Secs. 2.4 and 3.3.
[CrossRef]

1991 (1)

R. Freund and N. Nachtigal, “QMR: a quasi-minimal residual method for non-hermitian linear systems,” Numer. Math.60, 315–339 (1991).
[CrossRef]

1987 (2)

F. Kikuchi, “Mixed and penalty formulations for finite element analysis of an eigenvalue problem in electromagnetism,” Comput. Method Appl. Mech. Eng.64, 509–521 (1987).
[CrossRef]

A. Christ and H. L. Hartnagel, “Three-dimensional finite-difference method for the analysis of microwave-device embedding,” IEEE Trans. Microwave Theory Tech.35, 688–696 (1987).
[CrossRef]

1986 (2)

A. van der Sluis and H. van der Vorst, “The rate of convergence of conjugate gradients,” Numer. Math.48, 543–560 (1986).
[CrossRef]

Y. Saad and M. H. Schultz, “GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems,” SIAM J. Sci. Stat. Comp.7, 856–869 (1986).
[CrossRef]

1985 (1)

J. P. Webb, “The finite-element method for finding modes of dielectric-loaded cavities,” IEEE Trans. Microwave Theory Tech.33, 635–639 (1985).
[CrossRef]

1977 (1)

A. Jennings, “Influence of the eigenvalue spectrum on the convergence rate of the conjugate gradient method,” IMA J. Appl. Math.20, 61–72 (1977).
[CrossRef]

1972 (1)

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B6, 4370–4379 (1972).
[CrossRef]

Alumbaugh, D. L.

G. A. Newman and D. L. Alumbaugh, “Three-dimensional induction logging problems, Part 2: A finite-difference solution,” Geophys.67, 484–491 (2002).
[CrossRef]

Aruliah, D. A.

E. Haber, U. M. Ascher, D. A. Aruliah, and D. W. Oldenburg, “Fast simulation of 3D electromagnetic problems using potentials,” J. Comput. Phys.163, 150–171 (2000).
[CrossRef]

Ascher, U. M.

E. Haber, U. M. Ascher, D. A. Aruliah, and D. W. Oldenburg, “Fast simulation of 3D electromagnetic problems using potentials,” J. Comput. Phys.163, 150–171 (2000).
[CrossRef]

Ashcroft, N. W.

N. W. Ashcroft and N. D. Mermin, Solid State Physics, 1st ed. (SaundersCollege, 1976), Ch. 8.

Axelsson, O.

O. Axelsson, “Iteration number for the conjugate gradient method,” Math. Comput. Simulat.61, 421–435 (2003).
[CrossRef]

Beckermann, B.

B. Beckermann and A. B. J. Kuijlaars, “Superlinear convergence of conjugate gradients,” SIAM J. Numer. Anal.39, 300–329 (2001).
[CrossRef]

B. Beckermann and A. B. J. Kuijlaars, “On the sharpness of an asymptotic error estimate for conjugate gradients,” BIT Numer. Math.41, 856–867 (2001).
[CrossRef]

Beilenhoff, K.

K. Beilenhoff, W. Heinrich, and H. Hartnagel, “Improved finite-difference formulation in frequency domain for three-dimensional scattering problems,” IEEE Trans. Microwave Theory Tech.40, 540–546 (1992).
[CrossRef]

Benzi, M.

M. Benzi, G. H. Golub, and J. Liesen, “Numerical solution of saddle point problems,” Acta Numer.14, 1–137 (2005). Sec. 9.2.
[CrossRef]

Berryman, J. G.

N. J. Champagne, J. G. Berryman, and H. M. Buettner, “FDFD: A 3D finite-difference frequency-domain code for electromagnetic induction tomography,” J. Comput. Phys.170, 830–848 (2001).
[CrossRef]

Brongersma, M. L.

W. Cai, W. Shin, S. Fan, and M. L. Brongersma, “Elements for plasmonic nanocircuits with three-dimensional slot waveguides,” Adv. Mater.22, 5120–5124 (2010).
[CrossRef] [PubMed]

Buettner, H. M.

N. J. Champagne, J. G. Berryman, and H. M. Buettner, “FDFD: A 3D finite-difference frequency-domain code for electromagnetic induction tomography,” J. Comput. Phys.170, 830–848 (2001).
[CrossRef]

Cai, W.

W. Cai, W. Shin, S. Fan, and M. L. Brongersma, “Elements for plasmonic nanocircuits with three-dimensional slot waveguides,” Adv. Mater.22, 5120–5124 (2010).
[CrossRef] [PubMed]

Campbell, S. L.

S. L. Campbell, I. C. F. Ipsen, C. T. Kelley, and C. D. Meyer, “GMRES and the minimal polynomial,” BIT Numer. Math.36, 664–675 (1996).
[CrossRef]

Catrysse, P.

Champagne, N. J.

N. J. Champagne, J. G. Berryman, and H. M. Buettner, “FDFD: A 3D finite-difference frequency-domain code for electromagnetic induction tomography,” J. Comput. Phys.170, 830–848 (2001).
[CrossRef]

Christ, A.

A. Christ and H. L. Hartnagel, “Three-dimensional finite-difference method for the analysis of microwave-device embedding,” IEEE Trans. Microwave Theory Tech.35, 688–696 (1987).
[CrossRef]

Christy, R. W.

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B6, 4370–4379 (1972).
[CrossRef]

Druskin, V. L.

V. L. Druskin, L. A. Knizhnerman, and P. Lee, “New spectral Lanczos decomposition method for induction modeling in arbitrary 3-D geometry,” Geophys.64, 701–706 (1999).
[CrossRef]

Fan, S.

W. Shin and S. Fan, “Choice of the perfectly matched layer boundary condition for frequency-domain Maxwell’s equations solvers,” J. Comput. Phys.231, 3406–3431 (2012).
[CrossRef]

L. Verslegers, P. Catrysse, Z. Yu, W. Shin, Z. Ruan, and S. Fan, “Phase front design with metallic pillar arrays,” Opt. Lett.35, 844–846 (2010).
[CrossRef] [PubMed]

W. Cai, W. Shin, S. Fan, and M. L. Brongersma, “Elements for plasmonic nanocircuits with three-dimensional slot waveguides,” Adv. Mater.22, 5120–5124 (2010).
[CrossRef] [PubMed]

G. Veronis and S. Fan, “Bends and splitters in metal-dielectric-metal subwavelength plasmonic waveguides,” Appl. Phys. Lett.87, 131102–3 (2005).
[CrossRef]

G. Veronis and S. Fan, “Overview of simulation techniques for plasmonic devices,” in “Surface Plasmon Nanophotonics,” M. Brongersma and P. Kik, eds. (Springer, 2007), pp. 169–182.
[CrossRef]

Freund, R.

R. Freund and N. Nachtigal, “QMR: a quasi-minimal residual method for non-hermitian linear systems,” Numer. Math.60, 315–339 (1991).
[CrossRef]

Freund, R. W.

R. W. Freund, G. H. Golub, and N. M. Nachtigal, “Iterative solution of linear systems,” Acta Numer.1, 57–100 (1992). Secs. 2.4 and 3.3.
[CrossRef]

Golub, G. H.

M. Benzi, G. H. Golub, and J. Liesen, “Numerical solution of saddle point problems,” Acta Numer.14, 1–137 (2005). Sec. 9.2.
[CrossRef]

R. W. Freund, G. H. Golub, and N. M. Nachtigal, “Iterative solution of linear systems,” Acta Numer.1, 57–100 (1992). Secs. 2.4 and 3.3.
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics(Roberts & Company Publishers, 2005), 3rd ed. Sec. 2.3.2.

Goossens, S.

S. Goossens and D. Roose, “Ritz and harmonic Ritz values and the convergence of FOM and GMRES,” Numer. Linear Algebra Appl.6, 281–293 (1999).
[CrossRef]

Haber, E.

E. Haber, U. M. Ascher, D. A. Aruliah, and D. W. Oldenburg, “Fast simulation of 3D electromagnetic problems using potentials,” J. Comput. Phys.163, 150–171 (2000).
[CrossRef]

Hartnagel, H.

K. Beilenhoff, W. Heinrich, and H. Hartnagel, “Improved finite-difference formulation in frequency domain for three-dimensional scattering problems,” IEEE Trans. Microwave Theory Tech.40, 540–546 (1992).
[CrossRef]

Hartnagel, H. L.

A. Christ and H. L. Hartnagel, “Three-dimensional finite-difference method for the analysis of microwave-device embedding,” IEEE Trans. Microwave Theory Tech.35, 688–696 (1987).
[CrossRef]

Heinrich, W.

K. Beilenhoff, W. Heinrich, and H. Hartnagel, “Improved finite-difference formulation in frequency domain for three-dimensional scattering problems,” IEEE Trans. Microwave Theory Tech.40, 540–546 (1992).
[CrossRef]

Hiptmair, R.

R. Hiptmair, F. Kramer, and J. Ostrowski, “A robust Maxwell formulation for all frequencies,” IEEE Trans. Magn.44, 682–685 (2008).
[CrossRef]

Hou, J.

J. Hou, R. Mallan, and C. Torres-Verdin, “Finite-difference simulation of borehole EM measurements in 3D anisotropic media using coupled scalar-vector potentials,” Geophys.71, G225–G233 (2006).
[CrossRef]

Inan, U. S.

U. S. Inan and R. A. Marshall, Numerical Electromagnetics: The FDTD Method (Cambridge University, 2011). Ch. 14.
[CrossRef]

Ipsen, I. C. F.

S. L. Campbell, I. C. F. Ipsen, C. T. Kelley, and C. D. Meyer, “GMRES and the minimal polynomial,” BIT Numer. Math.36, 664–675 (1996).
[CrossRef]

Jennings, A.

A. Jennings, “Influence of the eigenvalue spectrum on the convergence rate of the conjugate gradient method,” IMA J. Appl. Math.20, 61–72 (1977).
[CrossRef]

Jin, J.-M.

J.-M. Jin, The Finite Element Method in Electromagnetics, 2nd ed. (Wiley, 2002).

Johnson, P. B.

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B6, 4370–4379 (1972).
[CrossRef]

Kelley, C. T.

S. L. Campbell, I. C. F. Ipsen, C. T. Kelley, and C. D. Meyer, “GMRES and the minimal polynomial,” BIT Numer. Math.36, 664–675 (1996).
[CrossRef]

Kikuchi, F.

F. Kikuchi, “Mixed and penalty formulations for finite element analysis of an eigenvalue problem in electromagnetism,” Comput. Method Appl. Mech. Eng.64, 509–521 (1987).
[CrossRef]

Knizhnerman, L. A.

V. L. Druskin, L. A. Knizhnerman, and P. Lee, “New spectral Lanczos decomposition method for induction modeling in arbitrary 3-D geometry,” Geophys.64, 701–706 (1999).
[CrossRef]

Koning, J. M.

D. A. White and J. M. Koning, “Computing solenoidal eigenmodes of the vector Helmholtz equation: a novel approach,” IEEE Trans. Magn.38, 3420–3425 (2002).
[CrossRef]

Kramer, F.

R. Hiptmair, F. Kramer, and J. Ostrowski, “A robust Maxwell formulation for all frequencies,” IEEE Trans. Magn.44, 682–685 (2008).
[CrossRef]

Kuijlaars, A. B. J.

B. Beckermann and A. B. J. Kuijlaars, “On the sharpness of an asymptotic error estimate for conjugate gradients,” BIT Numer. Math.41, 856–867 (2001).
[CrossRef]

B. Beckermann and A. B. J. Kuijlaars, “Superlinear convergence of conjugate gradients,” SIAM J. Numer. Anal.39, 300–329 (2001).
[CrossRef]

Lee, P.

V. L. Druskin, L. A. Knizhnerman, and P. Lee, “New spectral Lanczos decomposition method for induction modeling in arbitrary 3-D geometry,” Geophys.64, 701–706 (1999).
[CrossRef]

Liesen, J.

M. Benzi, G. H. Golub, and J. Liesen, “Numerical solution of saddle point problems,” Acta Numer.14, 1–137 (2005). Sec. 9.2.
[CrossRef]

Mallan, R.

J. Hou, R. Mallan, and C. Torres-Verdin, “Finite-difference simulation of borehole EM measurements in 3D anisotropic media using coupled scalar-vector potentials,” Geophys.71, G225–G233 (2006).
[CrossRef]

Marshall, R. A.

U. S. Inan and R. A. Marshall, Numerical Electromagnetics: The FDTD Method (Cambridge University, 2011). Ch. 14.
[CrossRef]

Mermin, N. D.

N. W. Ashcroft and N. D. Mermin, Solid State Physics, 1st ed. (SaundersCollege, 1976), Ch. 8.

Meyer, C. D.

S. L. Campbell, I. C. F. Ipsen, C. T. Kelley, and C. D. Meyer, “GMRES and the minimal polynomial,” BIT Numer. Math.36, 664–675 (1996).
[CrossRef]

Nachtigal, N.

R. Freund and N. Nachtigal, “QMR: a quasi-minimal residual method for non-hermitian linear systems,” Numer. Math.60, 315–339 (1991).
[CrossRef]

Nachtigal, N. M.

R. W. Freund, G. H. Golub, and N. M. Nachtigal, “Iterative solution of linear systems,” Acta Numer.1, 57–100 (1992). Secs. 2.4 and 3.3.
[CrossRef]

Newman, G. A.

C. J. Weiss and G. A. Newman, “Electromagnetic induction in a generalized 3D anisotropic earth, Part 2: The LIN preconditioner,” Geophys.68, 922–930 (2003).
[CrossRef]

G. A. Newman and D. L. Alumbaugh, “Three-dimensional induction logging problems, Part 2: A finite-difference solution,” Geophys.67, 484–491 (2002).
[CrossRef]

Oldenburg, D. W.

E. Haber, U. M. Ascher, D. A. Aruliah, and D. W. Oldenburg, “Fast simulation of 3D electromagnetic problems using potentials,” J. Comput. Phys.163, 150–171 (2000).
[CrossRef]

Ostrowski, J.

R. Hiptmair, F. Kramer, and J. Ostrowski, “A robust Maxwell formulation for all frequencies,” IEEE Trans. Magn.44, 682–685 (2008).
[CrossRef]

Roose, D.

S. Goossens and D. Roose, “Ritz and harmonic Ritz values and the convergence of FOM and GMRES,” Numer. Linear Algebra Appl.6, 281–293 (1999).
[CrossRef]

Ruan, Z.

Saad, Y.

Y. Saad and M. H. Schultz, “GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems,” SIAM J. Sci. Stat. Comp.7, 856–869 (1986).
[CrossRef]

Schultz, M. H.

Y. Saad and M. H. Schultz, “GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems,” SIAM J. Sci. Stat. Comp.7, 856–869 (1986).
[CrossRef]

Shin, W.

W. Shin and S. Fan, “Choice of the perfectly matched layer boundary condition for frequency-domain Maxwell’s equations solvers,” J. Comput. Phys.231, 3406–3431 (2012).
[CrossRef]

L. Verslegers, P. Catrysse, Z. Yu, W. Shin, Z. Ruan, and S. Fan, “Phase front design with metallic pillar arrays,” Opt. Lett.35, 844–846 (2010).
[CrossRef] [PubMed]

W. Cai, W. Shin, S. Fan, and M. L. Brongersma, “Elements for plasmonic nanocircuits with three-dimensional slot waveguides,” Adv. Mater.22, 5120–5124 (2010).
[CrossRef] [PubMed]

Simoncini, V.

V. Simoncini and D. B. Szyld, “Recent computational developments in Krylov subspace methods for linear systems,” Numer. Linear Algebra Appl.14, 1–59 (2007).
[CrossRef]

Smith, J. T.

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To obtain 8/Δmin2, take the first equation of Eq. (4.29) in [32] and then multiply the extra factor μ= μ0to account for the difference between Eq. (4.17a) in [32] and Eq. (1) in the present paper.

At the final stage of our work, we were made aware of a related work by M. Kordy, E. Cherkaev, and P. Wannamaker, “Schelkunoff potential for electromagnetic field: proof of existence and uniqueness” (to be published), where an equation similar to our Eq. (7) with s= −1 was developed.

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Figures (9)

Fig. 1
Fig. 1

A 2D square domain filled with vacuum (ε = ε0) for which the eigenvalue distribution of T is calculated numerically for s = 0, −1, +1. The domain is homogeneous in the z-direction, whereas its x- and y-boundaries are subject to periodic boundary conditions. The square domain is discretized on a finite-difference grid with cell size Δ = 2 nm. The domain is composed of 50 ×50 grid cells, which lead to 7500 eigenvalues in total. A vacuum wavelength λ0 = 1550 nm, which puts the system in the low-frequency regime, is assumed for the electric current source to be used in Sec. 3.

Fig. 2
Fig. 2

The eigenvalue distribution of A discretized from T for (a) s = 0, (b) s = −1, and (c) s = +1 for the vacuum-filled domain illustrated in Fig. 1. All 7500 eigenvalues λ of A are calculated for each s and categorized into 41 intervals in the horizontal axis that represents the range of the eigenvalues; the unit of the horizontal axis is nm−2. The height of the column on each interval represents the number of the eigenvalues in the interval. In (b) and (c), the black dots indicate the eigenvalue distribution for s = 0 shown in (a). The vertical axes are broken due to the extremely tall column at λ ≃ 0 in (a). The local maxima at λ = ±1 nm−2 are the Van Hove singularities [30] arising from the lattice structure imposed by the finite-difference grid.

Fig. 3
Fig. 3

Convergence behavior of GMRES for the vacuum-filled domain illustrated in Fig. 1. Three systems of linear equations discretized from Eq. (8) for s = 0, −1, +1 are solved by GMRES. In the iteration process of GMRES for each s, we plot the relative residual norm ||rm||/||b|| at each iteration step m. Notice that for s = 0 the relative residual norm stagnates initially; for s = −1 it stagnates around m = 100; for s = +1 it does not stagnate, but decreases very slowly. The upper and lower “X” marks on the vertical axis indicate the values around which our theory expects ||rm||/||b|| to stagnate for s = 0 and s = −1, respectively.

Fig. 4
Fig. 4

Initial evolution of rm/||b|| for s = −1. Relative residual vectors rm/||b|| are visualized at three iteration steps m = 0, 2, 4. In each plot, the column on each interval represents the norm of rm/||b|| projected onto the sum of the eigenspaces of the eigenvalues contained in the interval. Notice that all the columns almost vanish only after four iteration steps. In the plots for m = 2 and m = 4, the residual polynomials pm are also plotted as solid curves; note that they always satisfy the condition (18).

Fig. 5
Fig. 5

Initial evolution of rm/||b|| for s = 0. Relative residual vectors rm/||b|| are visualized at three iteration steps m = 0, 2, 4. In each plot, the column on each interval represents the norm of rm/||b|| projected onto the sum of the eigenspaces of the eigenvalues contained in the interval. Notice that most columns almost vanish only after four iteration steps, except for the very persistent column at λ ≃ 0. In the plots for m = 2 and m = 4, the residual polynomials pm are also plotted as solid curves; note that they always satisfy the condition (18).

Fig. 6
Fig. 6

Impact of the magnitude of the smallest root of a polynomial m ∈ ��m on the oscillation amplitudes of m. Three m of degree 6 are shown. In each figure, a solid line represents a polynomial; an open dot on the horizontal axis indicates the smallest root; solid dots indicate the other roots; dashed lines show the slopes of the polynomial at the roots. The three polynomials have the same roots except for their smallest roots: the smallest root in (a) becomes smaller positive and negative roots in (b) and (c), respectively. Notice that the slopes at all roots in (a) become steeper in (b) and (c) as the smallest root decreases in magnitude, and as a result the amplitudes of oscillation of m around the horizontal axis increase.

Fig. 7
Fig. 7

Evolution of rm/||b|| for s = +1. Relative residual vectors rm/||b|| are visualized at iteration steps m = 0, 4, 7 in the first row and at m = 11, 120, 140 in the second row. In each plot, the column on each interval represents the norm of rm/||b|| projected onto the sum of the eigenspaces of the eigenvalues contained in the interval. The vertical scale of the plot is magnified as the iteration proceeds. Notice that the column at λ ≃ 0 is very persistent during the later period of the iteration process (m = 120, 140). In the plots for m = 4, 7, 11, the residual polynomials pm are also plotted as solid curves; the residual polynomials are not plotted for m = 100 and m = 120 because they have too many roots.

Fig. 8
Fig. 8

Candidates for the residual polynomials for (a) a nearly positive-definite matrix and (b) strongly indefinite matrix. In each figure, a solid line represents a polynomial m ∈ ��m; an open dot on the horizontal axis indicates the smallest-magnitude root; solid dots indicate the other roots; dashed lines show the slopes of the polynomial at the roots. The two polynomials have the same smallest-magnitude root ζ0, and thus have approximately the same slope −1/ζ0 at their smallest-magnitude roots. Note that for both m the slopes get steeper at the roots further away from the median of the roots. Hence, the slopes of most dashed lines are gentler than 1/|ζ0| in (a) and steeper than 1/|ζ0| in (b). As a result, m in (b) has larger amplitudes of oscillation around the horizontal axis than m in (a).

Fig. 9
Fig. 9

Three inhomogeneous systems for which Eq. (7) is solved for s = −1, 0, +1 by QMR: (a) a slot waveguide bend formed in a thin silver film (Slot), (b) a straight silicon waveguide (Diel), and (c) an array of gold pillars (Array). The figures in the first row describe the three systems. The directions of wave propagation are shown by red arrows, beside which the vacuum wavelengths used are indicated. For all three systems, the waves are excited by electric current sources J strictly inside the simulation domain. The plots in the second row show the convergence behavior of QMR. Note that for all three systems QMR converges fastest for s = −1, whereas it barely converges for s = +1. The relative electric permittivities of the materials used in these systems are ε r silver = 129 i 3.28 [34], ε r silica = 2.085 [35], ε r silicon = 12.09 [35], and ε r gold = 10.78 i 0.79 [36], respectively.

Tables (3)

Tables Icon

Table 1 Properties of the eigenvalue distributions of T0 for different s. Depending on the sign of s, T0 has very different eigenvalue distributions in terms of the multiplicity of the eigenvalue 0 and the definiteness of T0.

Tables Icon

Table 2 Specification of the finite-difference grids used for the three systems in Fig. 9. Slot uses a nonuniform grid with smoothly varying grid cell size. The matrix A has 3NxNyNz rows and columns, where the extra factor 3 accounts for the three Cartesian components of the E-field.

Tables Icon

Table 3 Parameters used in Eq. (16) for the three systems in Fig. 9. When substituted in Eq. (16), these parameters prove that all the three systems are in the low-frequency regime.

Equations (45)

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× × E ω 2 μ 0 ε E = i ω μ 0 J ,
A x = b ,
2 Ψ ω 2 μ 0 ε ( Ψ + φ ) = i ω μ 0 J ,
( ε E ) = 0 .
× × E + s [ ( ( ε / ε 0 ) E ) ] ω 2 μ 0 ε E = 0
i ω ρ + J = 0 , or ( ε E ) = i ω J ,
× × E + s [ ε 1 ( ε E ) ] ω 2 μ 0 ε E = i ω μ 0 J + s i ω [ ε 1 J ]
× × E + s ( E ) ω 2 μ 0 ε E = i ω μ 0 J + s i ω ε ( J ) ,
T = × ( × ) + s ( ) ω 2 μ 0 ε
T 0 = × ( × ) + s ( )
F k = [ k x k y k z ] , [ k z 0 k x ] , [ k y k x 0 ]
λ = 0 , | k | 2 , | k | 2
λ = | k | 2 , 0 , 0
λ = s | k | 2 , | k | 2 , | k | 2
ω 2 μ 0 | ε | 8 / Δ min 2 .
λ 0 / Δ min π | ε r | / 2 ,
r m = b A x m
p ˜ m ( 0 ) = 1 .
r ˜ m p ˜ m ( A ) r 0 .
r m = p m ( A ) r 0 .
A = V Λ V ,
Λ = [ λ 1 λ n ] , V = [ v 1 v n ]
r ˜ m = V p ˜ m ( Λ ) V r 0
z ˜ m V ( r ˜ m / b ) ,
z m V ( r m / b ) .
z ˜ m = p ˜ m ( Λ ) z 0 = [ p ˜ m ( λ 1 ) p ˜ m ( λ n ) ] z 0 ,
z ˜ m i = p ˜ m ( λ i ) z 0 i .
h m j = [ λ i t j z m i 2 ] 1 / 2 .
× μ 1 × E + s [ ( μ ε ) 1 ( ε E ) ] ω 2 ε E = i ω J + s i ω [ ( μ ε ) 1 J ] ,
F = F k e i k r ,
[ k y 2 + k z 2 k x k y k x k z k y k x k z 2 + k x 2 k y k z k z k x k z k y k x 2 + k y 2 ] [ F k x F k y F k z ] = λ [ F k x F k y F k z ] ,
[ k x 2 k x k y k x k z k y k x k y 2 k y k z k z k x k z k y k z 2 ] [ F k x F k y F k z ] = λ [ F k x F k y F k z ] .
λ = 0 , | k | 2 , | k | 2 ,
λ = | k | 2 , 0 , 0 ,
F k = [ k x k y k z ] , [ k z 0 k x ] , [ k y k x 0 ]
p ˜ m ( ζ ) = i = 1 d m ( 1 ζ ζ i ) ,
p ˜ m ' ( ζ k ) = 1 ζ k i k ( 1 ζ k ζ i ) .
| p ˜ m ' ( ζ 1 ) | = 1 ζ 1 ( 1 ζ 1 ζ 2 ) ( 1 ζ 1 ζ d m ) ,
| p ˜ m ' ( ζ k ) | = ( ζ k ζ 1 1 ) [ 1 ζ k i 1 , k | 1 ζ k ζ i | ] ,
| p ˜ m ' ( ζ 1 ) | = 1 | ζ 1 | ( 1 + | ζ 1 | ζ 2 ) ( 1 + | ζ 1 | ζ d m )
| p ˜ m ' ( ζ k ) | = ( ζ k | ζ 1 | + 1 ) [ 1 ζ k i 1 , k | 1 ζ k ζ i | ]
p ( ζ ) = α i = 1 m ( ζ ζ i ) ,
p ( ζ k ) = α i k ( ζ k ζ i ) .
| p ( ζ k + 1 ) | | p ( ζ k ) | = k ! ( m k 1 ) ! ( k 1 ) ! ( m k ) ! = k m k .
| p ( ζ k + 1 ) | | p ( ζ k ) | = i k + 1 | ζ k + 1 ζ i | i k | ζ k ζ i | = i k , k + 1 | ζ k + 1 ζ i | | ζ k ζ i | = i = 1 k 1 ( ζ k + 1 ζ i ζ k ζ i ) i = k + 2 m ( ζ i ζ k + 1 ζ i ζ k ) = [ i = 1 k 1 ( 1 + ζ k + 1 ζ k ζ k ζ i ) ] [ i = k + 2 m ( 1 ζ k + 1 ζ k ζ i ζ k ) ] .

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