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Choice of the perfectly matched layer boundary condition for frequency-domain Maxwell’s equations solvers. (English) Zbl 1242.78035
Summary: We show that the performance of frequency-domain solvers for Maxwell’s equations is greatly affected by the kind of perfectly matched layer (PML) used. In particular, we demonstrate that using the stretched-coordinate PML results in significantly faster convergence speed than using the uniaxial PML (UPML). Such a difference in convergence behavior is explained by an analysis of the condition number of the coefficient matrices. Additionally, we develop a diagonal preconditioning scheme that significantly improves solver performance when UPML is used.

MSC:
78M20 Finite difference methods applied to problems in optics and electromagnetic theory
78M10 Finite element, Galerkin and related methods applied to problems in optics and electromagnetic theory
65F08 Preconditioners for iterative methods
65F15 Numerical computation of eigenvalues and eigenvectors of matrices
78A50 Antennas, waveguides in optics and electromagnetic theory
65F10 Iterative numerical methods for linear systems
35Q61 Maxwell equations
Software:
ARPACK; FDFD; Matlab
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References:
[1] Bérenger, J.-P., A perfectly matched layer for the absorption of electromagnetic waves, Journal of computational physics, 114, 185-200, (1994) · Zbl 0814.65129
[2] Veronis, G.; Fan, S., Theoretical investigation of compact couplers between dielectric slab waveguides and two-dimensional metal-dielectric-metal plasmonic waveguides, Optics express, 15, 1211-1221, (2007)
[3] Verslegers, L.; Catrysse, P.; Yu, Z.; Shin, W.; Ruan, Z.; Fan, S., Phase front design with metallic pillar arrays, Optics letters, 35, 844-846, (2010)
[4] Taflove, A.; Hagness, S.C., Computational electrodynamics: the finite-difference time-domain method, (2005), Artech House Publishers
[5] Sacks, Z.; Kingsland, D.; Lee, R.; Lee, J.-F., A perfectly matched anisotropic absorber for use as an absorbing boundary condition, IEEE transactions on antennas and propagation, 43, 1460-1463, (1995)
[6] Chew, W.C.; Weedon, W.H., A 3D perfectly matched medium from modified maxwell’s equations with stretched coordinates, Microwave and optical technology letters, 7, 599-604, (1994)
[7] Rappaport, C.M., Perfectly matched absorbing boundary conditions based on anisotropic lossy mapping of space, Microwave and guided wave letters, IEEE, 5, 90-92, (1995)
[8] Mittra, R.; Pekel, U., A new look at the perfectly matched layer (PML) concept for the reflectionless absorption of electromagnetic waves, Microwave and guided wave letters, IEEE, 5, 84-86, (1995)
[9] Roden, J.; Gedney, S., Convolution PML (CPML): an efficient FDTD implementation of the CFS-PML for arbitrary media, Microwave and optical technology letters, 27, 334-339, (2000)
[10] Wu, J.-Y.; Kingsland, D.; Lee, J.-F.; Lee, R., A comparison of anisotropic PML to berenger’s PML and its application to the finite-element method for EM scattering, IEEE transactions on antennas and propagation, 45, 40-50, (1997)
[11] Y. Botros, J. Volakis, A robust iterative scheme for FEM applications terminated by the perfectly matched layer (PML) absorbers, Proceedings of the Fifteenth National Radio Science Conference, 1998, pp. D11/1-D11/8.
[12] Stupfel, B., A study of the condition number of various finite element matrices involved in the numerical solution of maxwell’s equations, IEEE transactions on antennas and propagation, 52, 3048-3059, (2004) · Zbl 1368.78147
[13] P. Talukder, F.-J. Schmuckle, R. Schlundt, W. Heinrich, Optimizing the FDFD method in order to minimize PML-related numerical problems, in: 2007 International Microwave Symposium (IMS 2007), 2007, pp. 293-296.
[14] Botros, Y.; Volakis, J., Preconditioned generalized minimal residual iterative scheme for perfectly matched layer terminated applications, Microwave and guided wave letters, IEEE, 9, 45-47, (1999)
[15] Jin, J.-M.; Chew, W., Combining PML and ABC for the finite-element analysis of scattering problems, Microwave and optical technology letters, 12, 192-197, (1996)
[16] Yee, K., Numerical solution of initial boundary value problems involving maxwell’s equations in isotropic media, IEEE transactions on antennas and propagation, 14, 302-307, (1966) · Zbl 1155.78304
[17] Smith, J., Conservative modeling of 3-D electromagnetic fields, part I: properties and error analysis, Geophysics, 61, 1308-1318, (1996)
[18] Champagne, N.J.; Berryman, J.; Buettner, H., FDFD: A 3D finite-difference frequency-domain code for electromagnetic induction tomography, Journal of computational physics, 170, 830-848, (2001) · Zbl 0984.78012
[19] Kunz, K.S.; Luebbers, R.J., The finite difference time domain method for electromagnetics, (1993), CRC-Press, Section 3.2
[20] Veronis, G.; Fan, S., Modes of subwavelength plasmonic slot waveguides, Journal of lightwave technology, 25, 2511-2521, (2007), In the private communication with the authors, the use of the 1nm grid edge length in this paper was confirmed
[21] Johnson, P.B.; Christy, R.W., Optical constants of the noble metals, Physical review B, 6, 4370-4379, (1972)
[22] ()
[23] ()
[24] Freund, R.; Nachtigal, N., QMR: a quasi-minimal residual method for non-Hermitian linear systems, Numerische Mathematik, 60, 315-339, (1991) · Zbl 0754.65034
[25] S. Balay, J. Brown, K. Buschelman, W.D. Gropp, D. Kaushik, M.G. Knepley, L.C. McInnes, B.F. Smith, H. Zhang, PETSc Web page, 2011. Available from: <http://www.mcs.anl.gov/petsc>.
[26] Jacobs, D.A.H., A generalization of the conjugate-gradient method to solve complex systems, IMA journal of numerical analysis, 6, 447-452, (1986) · Zbl 0614.65028
[27] Benzi, M.; Golub, G.H.; Liesen, J., Numerical solution of saddle point problems, Acta numerica, 14, 1-137, (2005), Section 9.2 · Zbl 1115.65034
[28] Datta, B.N., Numerical linear algebra and applications, (2010), SIAM, Section 6.8
[29] Golub, G.H.; Van Loan, C.F., Matrix computations, (1996), The Johns Hopkins University Press, Section 2.5.6; 2.3.1; 2.3.3 · Zbl 0865.65009
[30] R.B. Lehoucq, K. Maschhoff, D.C. Sorensen, C. Yang, ARPACK Web page, 2011. Available from: <http://www.caam.rice.edu/software/ARPACK>.
[31] MATLAB Web page, 2011. Available from: <http://www.mathworks.com/products/matlab>.
[32] Lehoucq, R.B.; Sorensen, D.C.; Yang, C., ARPACK users’ guide: solution of large-scale eigenvalue problems with implicitly restarted Arnoldi methods, (1998), SIAM · Zbl 0901.65021
[33] Goodman, J.W., Introduction to Fourier optics, (2005), Roberts & Company Publishers, Section 2.3.2
[34] Oppenheim, A.V.; Schafer, R.W.; Buck, J.R., Discrete-time signal processing, (1999), Prentice Hall, Section 2.6.1; 4.2
[35] Wolfe, C.; Navsariwala, U.; Gedney, S., A parallel finite-element tearing and interconnecting algorithm for solution of the vector wave equation with PML absorbing medium, IEEE transactions on antennas and propagation, 48, 278-284, (2000)
[36] Kottke, C.; Farjadpour, A.; Johnson, S., Perturbation theory for anisotropic dielectric interfaces, and application to subpixel smoothing of discretized numerical methods, Physical review E, 77, 036611, (2008), Appendix
[37] Teixeira, F.; Chew, W., General closed-form PML constitutive tensors to match arbitrary bianisotropic and dispersive linear media, Microwave and guided wave letters, IEEE, 8, 223-225, (1998)
[38] Gedney, S., An anisotropic perfectly matched layer-absorbing medium for the truncation of FDTD lattices, IEEE transactions on antennas and propagation, 44, 1630-1639, (1996)
[39] Landau, L.D.; Lifshitz, E.M., Quantum mechanics: non-relativistic theory, Course of theoretical physics, vol. 3, (1977), Butterworth-Heinemann · Zbl 0178.57901
[40] Takagi, T., On an algebraic problem related to an analytic theorem of carathedory and fejer and on an allied theorem of Landau, Japanese journal of mathematics, 1, 82-93, (1924)
[41] Horn, R.A.; Johnson, C.R., Matrix analysis, (1985), Cambridge University Press, Corollary 4.4.4; Theorem 7.3.5 · Zbl 0576.15001
[42] Bunse-Gerstner, A.; Gragg, W., Singular value decompositions of complex symmetric matrices, Journal of computational and applied mathematics, 21, 41-54, (1988) · Zbl 0635.65031
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