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Mathematical analysis of Ziolkowski’s PML model with application for wave propagation in metamaterials. (English) Zbl 1431.78010
Summary: In this paper, we investigate one perfectly matched layer (PML) model proposed by R. W. Ziolkowski [Comput. Methods Appl. Mech. Eng. 169, No. 3–4, 237–262 (1999; Zbl 0960.78018)]. Various schemes for solving this PML model have been developed and have been shown to be effective in absorbing outgoing waves when the wave propagation in unbounded domain problem is reduced to a bounded domain problem. However, a rigorous analysis of this model is lacking. In this paper we establish the stability of this PML model and propose a fully-discrete finite element scheme to solve this model with edge elements. Discrete stability and optimal error estimate of this scheme are proved. Numerical results justifying the analysis and demonstrating the effectiveness of this PML model are presented.
78M10 Finite element, Galerkin and related methods applied to problems in optics and electromagnetic theory
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35L15 Initial value problems for second-order hyperbolic equations
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
78A25 Electromagnetic theory, general
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
FEniCS; SyFi
Full Text: DOI
[1] Bérenger, J.-P., A perfectly matched layer for the absorption of electromagnetic waves, J. Comput. Phys., 114, 2, 185-200 (1994) · Zbl 0814.65129
[2] Bao, G.; Li, P.; Wu, H., An adaptive edge element method with perfectly matched absorbing layers for wave scattering by biperiodic structures, Math. Comp., 79, 1-34 (2010) · Zbl 1197.78031
[3] Bokil, V. A.; Buksas, M. W., Comparison of finite difference and mixed finite element methods for perfectly matched layer models, Commun. Comput. Phys., 2, 806-826 (2007)
[4] Huang, Y.; Jia, H.; Li, J., Analysis and application of an equivalent Berenger’s PML model, J. Comput. Appl. Math., 333, 157-169 (2018) · Zbl 1382.78018
[5] Huang, Y.; Li, J.; Yang, W., Mathematical analysis of a PML model obtained with stretched coordinates and its application to backward wave propagation in metamaterials, Numer. Methods Partial Differential Equations, 30, 1558-1574 (2014) · Zbl 1308.78023
[6] Lin, Y.; Zhang, K.; Zou, J., Studies on some perfectly matched layers for one-dimensional time-dependent systems, Adv. Comput. Math., 30, 1-35 (2009) · Zbl 1166.65042
[7] Lu, T.; Zhang, P.; Cai, W., Discontinuous Galerkin methods for dispersive and lossy Maxwell’s equations and PML boundary conditions, J. Comput. Phys., 200, 549-580 (2004) · Zbl 1115.78330
[8] Turkel, E.; Yefet, A., Absorbing PML boundary layers for wave-like equations, Appl. Numer. Math., 27, 533-557 (1998) · Zbl 0933.35188
[9] Appelö, D.; Hagstrom, T.; Kreiss, G., Perfectly matched layers for hyperbolic systems: general formulation, well-posedness, and stability, SIAM J. Appl. Math., 67, 1, 1-23 (2006) · Zbl 1110.35042
[10] Diaz, J.; Joly, P., A time domain analysis of PML models in acoustics, Comput. Methods Appl. Mech. Engrg., 195, 3820-3853 (2006) · Zbl 1119.76046
[11] Bécache, E.; Joly, P.; Kachanovska, M.; Vinoles, V., Perfectly matched layers in negative index metamaterials and plasmas, ESAIM Proc. Surv., 50, 113-132 (2015) · Zbl 1341.78013
[12] Bonnet-Ben Dhia, A.-S.; Carvalho, C.; Chesnel, L.; Ciarlet Jr, P., On the use of perfectly matched layers at corners for scattering problems with sign-changing coefficients, J. Comput. Phys., 322, 224-247 (2016) · Zbl 1351.78035
[13] Taflove, A.; Hagness, S. C., Computational Electrodynamics: The Finite-Difference Time-Domain Method (2000), Artech House · Zbl 0963.78001
[14] Li, J.; Huang, Y., (Time-Domain Finite Element Methods for Maxwell’s Equations in Metamaterials. Time-Domain Finite Element Methods for Maxwell’s Equations in Metamaterials, Springer Series in Computational Mathematics, vol. 43 (2013), Springer) · Zbl 1304.78002
[15] Huang, Y.; Li, J., Numerical analysis of a PML model for time-dependent Maxwell’s equations, J. Comput. Appl. Math., 235, 3932-3942 (2011) · Zbl 1220.78112
[16] Ziolkowski, R. W., Maxwellian material based absorbing boundary conditions, Comput. Methods Appl. Mech. Engrg., 169, 237-262 (1999) · Zbl 0960.78018
[17] Brenner, S. C.; Gedicke, J.; Sung, L.-Y., An adaptive \(P_1\) finite element method for two-dimensional transverse magnetic time harmonic Maxwell’s equations with general material properties and general boundary conditions, J. Sci. Comput., 68, 2, 848-863 (2016) · Zbl 1373.78416
[18] Chung, E. T.; Ciarlet Jr, P., A staggered discontinuous Galerkin method for wave propagation in media with dielectrics and meta-materials, J. Comput. Appl. Math., 239, 189-207 (2013) · Zbl 1260.78007
[19] Demkowicz, L.; Li, J., Numerical simulations of cloaking problems using a DPG method, Comput. Mech., 51, 661-672 (2013) · Zbl 1311.78011
[20] Li, W.; Liang, D.; Lin, Y., A new energy-conserved S-FDTD scheme for Maxwell’s equations in metamaterials, Int. J. Numer. Anal. Model., 10, 775-794 (2013) · Zbl 1277.78033
[21] Li, J.; Shi, C.; Shu, C.-W., Optimal non-dissipative discontinuous Galerkin methods for Maxwell’s equations in Drude metamaterials, Comput. Math. Appl., 73, 1760-1780 (2017) · Zbl 1370.74143
[22] Yang, Z.; Wang, L.-L.; Rong, Z.; Wang, B.; Zhang, B., Seamless integration of global Dirichlet-to-Neumann boundary condition and spectral elements for transformation electromagnetics, Comput. Methods Appl. Mech. Engrg., 301, 137-163 (2016) · Zbl 1423.78008
[23] Li, J., Development of discontinuous Galerkin methods for Maxwell’s equations in metamaterials and perfectly matched layers, J. Comput. Appl. Math., 236, 950-961 (2011) · Zbl 1252.78032
[24] Li, J.; Hesthaven, J. S., Analysis and application of the nodal discontinuous Galerkin method for wave propagation in metamaterials, J. Comput. Phys., 258, 915-930 (2014) · Zbl 1349.74327
[25] (Logg, A.; Mardal, K.-A.; Wells, G. N., Automated Solution of Differential Equations by the Finite Element Method: The FEniCS Book (2012), Springer) · Zbl 1247.65105
[26] Monk, P., Finite Element Methods for Maxwell’s Equations (2003), Oxford University Press · Zbl 1024.78009
[27] Huang, Y.; Li, J.; Wu, C., Averaging for superconvergence: verification and application of 2D edge elements to Maxwell’s equations in metamaterials, Comput. Methods Appl. Mech. Engrg., 255, 121-132 (2013) · Zbl 1297.78016
[28] Ziolkowski, R. W., Pulsed and CW Gaussian beam interactions with double negative metamaterial slabs, Opt. Express, 11, 662-681 (2003)
[29] Huang, Y.; Li, J.; Yang, W., Modeling backward wave propagation in metamaterials by the finite element time domain method, SIAM J. Sci. Comput., 35, B248-B274 (2013) · Zbl 1274.78090
[30] Li, J.; Wood, A., Finite element analysis for wave propagation in double negative metamaterials, J. Sci. Comput., 32, 263-286 (2007) · Zbl 1143.78367
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