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On the mod p Steenrod algebra and the Leibniz-Hopf algebra
An adaptive edge finite element method for the Maxwell's equations in metamaterials
1. | Hunan Key Laboratory for Computation and Simulation in Science and Engineering, School of Mathematics and Computational Science, Xiangtan University, Xiangtan 411105, Hunan, China |
2. | Key Laboratory of Intelligent Computing & Information Processing of Ministry of Education, School of Mathematics and Computational Science, Xiangtan University, Xiangtan 411105, Hunan, China |
In this paper, we study an adaptive edge finite element method for time-harmonic Maxwell's equations in metamaterials. A-posteriori error estimators based on the recovery type and residual type are proposed, respectively. Based on our a-posteriori error estimators, the adaptive edge finite element method is designed and applied to simulate the backward wave propagation, electromagnetic splitter, rotator, concentrator and cloak devices. Numerical examples are presented to illustrate the reliability and efficiency of the proposed a-posteriori error estimations for the adaptive method.
References:
[1] |
K. Ando, H. Kang and H. Liu,
Plasmon resonance with finite frequencies: A validation of the quasi-static approximation for diametrically small inclusions, SIAM J. Appl. Math., 76 (2016), 731-749.
doi: 10.1137/15M1025943. |
[2] |
G. Bao, H. Liu and J. Zou,
Nearly cloaking the full Maxwell equations: Cloaking active contents with general conducting layers, J. Math. Pures. Appl. (9), 101 (2014), 716-733.
doi: 10.1016/j.matpur.2013.10.010. |
[3] |
J.-P. Berenger,
A perfectly matched layer for the absorption of electromagnetic waves, J. Comput. Phys., 114 (1994), 185-200.
doi: 10.1006/jcph.1994.1159. |
[4] |
E. Blåsten, H. Li, H. Liu and Y. Wang,
Localization and geometrization in plasmon resonances and geometric structures of Neumann-Poincaré eigenfunctions, ESAIM Math. Model. Numer. Anal., 54 (2020), 957-976.
doi: 10.1051/m2an/2019091. |
[5] |
J. H. Bramble and J. E. Pasciak,
Analysis of a Cartesian PML approximation to the three dimensional electromagnetic wave scattering problem, Int. J. Numer. Anal. Model., 9 (2012), 543-561.
|
[6] |
S. C. Brenner, J. Gedicke and L.-Y. Sung,
An adaptive $P_1$ finite element method for two-dimensional Maxwell's equations, J. Sci. Comput., 55 (2013), 738-754.
doi: 10.1007/s10915-012-9658-8. |
[7] |
S. C. Brenner, J. Gedicke and L.-Y. Sung,
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 (2016), 848-863.
doi: 10.1007/s10915-015-0161-x. |
[8] |
Z. Cai and S. Cao,
A recovery-based a posteriori error estimator for $H$(curl) interface problems, Comput. Methods. Appl. Mech. Engrg., 296 (2015), 169-195.
doi: 10.1016/j.cma.2015.08.002. |
[9] |
Z. Cai and S. Zhang,
Recovery-based error estimators for interface problems: Mixed and nonconforming finite elements, SIAM J. Numer. Anal., 48 (2010), 30-52.
doi: 10.1137/080722631. |
[10] |
Z. Cai and S. Zhang,
Flux recovery and a posteriori error estimators: Conforming elements for scalar elliptic equations, SIAM J. Numer. Anal., 48 (2010), 578-602.
doi: 10.1137/080742993. |
[11] |
J. Cui,
Multigrid methods for two-dimensional Maxwell's equations on graded meshes, J. Comput. Appl. Math., 255 (2014), 231-247.
doi: 10.1016/j.cam.2013.05.007. |
[12] |
Y. Deng, H. Liu and G. Uhlmann,
Full and partial cloaking in electromagnetic scattering, Arch. Ration. Mech. Anal., 223 (2017), 265-299.
doi: 10.1007/s00205-016-1035-6. |
[13] |
W. Dörfler,
A convergent adaptive algorithm for Poisson's equation, SIAM J. Numer. Anal., 33 (1996), 1106-1124.
doi: 10.1137/0733054. |
[14] |
Y. Hao and R. Mittra, FDTD modeling of metamaterials: Theory and applications, Artech. House., (2008). Google Scholar |
[15] |
B. He, W. Yang and H. Wang, Convergence analysis of adaptive edge finite element method for variable coefficient time-harmonic Maxwell's equations, J. Comput. Appl. Math., 376 (2020), 16pp.
doi: 10.1016/j.cam.2020.112860. |
[16] |
Y. Huang, J. Li and C. Wu,
The averaging technique for superconvergence: Verification and application of 2D edge elements to Maxwell's equations in metamaterials, Comput. Methods Appl. Mech. Engrg., 255 (2013), 121-132.
doi: 10.1016/j.cma.2012.11.008. |
[17] |
Y. Huang, J. Li and W. Yang,
Interior penalty DG methods for Maxwell's equations in dispersive media, J. Comput. Phys., 230 (2011), 4559-4570.
doi: 10.1016/j.jcp.2011.02.031. |
[18] |
Y. Huang, J. Li and W. Yang, Modeling backward wave propagation in metamaterials by the finite element time-domain method, SIAM J. Sci. Comput., 35 (2013), B248–B274.
doi: 10.1137/120869869. |
[19] |
Y. Huang and N. Yi,
The superconvergent cluster recovery method, J. Sci. Comput., 44 (2010), 301-322.
doi: 10.1007/s10915-010-9379-9. |
[20] |
H. Li, J. Li and H. Liu,
On quasi-static cloaking due to anomalous localized resonance in $\mathbb{R}^{3}$, SIAM J. Appl. Math., 75 (2015), 1245-1260.
doi: 10.1137/15M1009974. |
[21] |
H. Li, S. Li, H. Liu and X. Wang,
Analysis of electromagnetic scattering from plasmonic inclusions beyond the quasi-static approximation and applications, ESAIM Math. Model. Numer. Anal., 53 (2019), 1351-1371.
doi: 10.1051/m2an/2019004. |
[22] |
J. Li and J. S. Hesthaven,
Analysis and application of the nodal discontinuous Galerkin method for wave propagation in metamaterials, J. Comput. Phys., 258 (2014), 915-930.
doi: 10.1016/j.jcp.2013.11.018. |
[23] |
J. Li, Y. Huang and W. Yang,
An adaptive edge finite element method for electromagnetic cloaking simulation, J. Comput. Phys., 249 (2013), 216-232.
doi: 10.1016/j.jcp.2013.04.026. |
[24] |
H. Liu, Virtual reshaping and invisibility in obstacle scattering, Inverse Problems, 25 (2009), 16pp.
doi: 10.1088/0266-5611/25/4/045006. |
[25] |
P. Monk, Finite Element Methods for Maxwell's Equations, Oxford University Press, New York, 2003.
doi: 10.1093/acprof:oso/9780198508885.001.0001.![]() ![]() |
[26] |
A. Naga and Z. Zhang,
A posteriori error estimates based on the polynomial preserving recovery, SIAM J. Numer. Anal., 42 (2004), 1780-1800.
doi: 10.1137/S0036142903413002. |
[27] |
N. C. Nguyen, J. Peraire and B. Cockburn,
Hybridizable discontinuous Galerkin methods for the time-harmonic Maxwell's equations, J. Comput. Phys., 230 (2011), 7151-7175.
doi: 10.1016/j.jcp.2011.05.018. |
[28] |
J. B. Pendry, D. Schurig and D. R. Smith,
Controlling electromagnetic fields, Science, 312 (2006), 1780-1782.
doi: 10.1126/science.1125907. |
[29] |
N. A. Pierce and M. B. Giles,
Adjoint recovery of superconvergent functionals from PDE approximations, SIAM. Rev., 42 (2000), 247-264.
doi: 10.1137/S0036144598349423. |
[30] |
A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, Artech House, Inc., Boston, MA, 2000. |
[31] |
H. Wang, W. Yang and Y. Huang,
Adaptive finite element method for the sound wave problems in two kinds of media, Comput. Math. Appl., 79 (2020), 789-801.
doi: 10.1016/j.camwa.2019.07.029. |
[32] |
D. H. Werner and D.-H. Kwon, Transformation Electromagnetics and Metamaterials. Fundamental Principles and Applications, Springer-Verlag, London, 2014.
doi: 10.1007/978-1-4471-4996-5. |
[33] |
W. Yang, Y. Huang and J. Li,
Developing a time-domain finite element method for the Lorentz metamaterial model and applications, J. Sci. Comput., 68 (2016), 438-463.
doi: 10.1007/s10915-015-0144-y. |
[34] |
W. Yang, J. Li and Y. Huang,
Modeling and analysis of the optical black hole in metamaterials by the finite element time-domain method, Comput. Methods Appl. Mech. Engrg., 304 (2016), 501-520.
doi: 10.1016/j.cma.2016.02.029. |
[35] |
W. Yang, J. Li and Y. Huang,
Mathematical analysis and finite element time domain simulation of arbitrary star-shaped electromagnetic cloaks, SIAM J. Numer. Anal., 56 (2018), 136-159.
doi: 10.1137/16M1093835. |
[36] |
W. Yang, J. Li, Y. Huang and B. He,
Developing finite element methods for simulating transformation optics devices with metamaterials, Commun. Comput. Phys., 25 (2019), 135-154.
doi: 10.4208/cicp.oa-2017-0225. |
[37] |
O. C. Zienkiewicz and J. Z. Zhu,
The superconvergence patch recovery (SPR) and adaptive finite element refinement, Comput. Methods. Appl. Mech. Engrg., 101 (1992), 207-224.
doi: 10.1016/0045-7825(92)90023-D. |
show all references
References:
[1] |
K. Ando, H. Kang and H. Liu,
Plasmon resonance with finite frequencies: A validation of the quasi-static approximation for diametrically small inclusions, SIAM J. Appl. Math., 76 (2016), 731-749.
doi: 10.1137/15M1025943. |
[2] |
G. Bao, H. Liu and J. Zou,
Nearly cloaking the full Maxwell equations: Cloaking active contents with general conducting layers, J. Math. Pures. Appl. (9), 101 (2014), 716-733.
doi: 10.1016/j.matpur.2013.10.010. |
[3] |
J.-P. Berenger,
A perfectly matched layer for the absorption of electromagnetic waves, J. Comput. Phys., 114 (1994), 185-200.
doi: 10.1006/jcph.1994.1159. |
[4] |
E. Blåsten, H. Li, H. Liu and Y. Wang,
Localization and geometrization in plasmon resonances and geometric structures of Neumann-Poincaré eigenfunctions, ESAIM Math. Model. Numer. Anal., 54 (2020), 957-976.
doi: 10.1051/m2an/2019091. |
[5] |
J. H. Bramble and J. E. Pasciak,
Analysis of a Cartesian PML approximation to the three dimensional electromagnetic wave scattering problem, Int. J. Numer. Anal. Model., 9 (2012), 543-561.
|
[6] |
S. C. Brenner, J. Gedicke and L.-Y. Sung,
An adaptive $P_1$ finite element method for two-dimensional Maxwell's equations, J. Sci. Comput., 55 (2013), 738-754.
doi: 10.1007/s10915-012-9658-8. |
[7] |
S. C. Brenner, J. Gedicke and L.-Y. Sung,
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 (2016), 848-863.
doi: 10.1007/s10915-015-0161-x. |
[8] |
Z. Cai and S. Cao,
A recovery-based a posteriori error estimator for $H$(curl) interface problems, Comput. Methods. Appl. Mech. Engrg., 296 (2015), 169-195.
doi: 10.1016/j.cma.2015.08.002. |
[9] |
Z. Cai and S. Zhang,
Recovery-based error estimators for interface problems: Mixed and nonconforming finite elements, SIAM J. Numer. Anal., 48 (2010), 30-52.
doi: 10.1137/080722631. |
[10] |
Z. Cai and S. Zhang,
Flux recovery and a posteriori error estimators: Conforming elements for scalar elliptic equations, SIAM J. Numer. Anal., 48 (2010), 578-602.
doi: 10.1137/080742993. |
[11] |
J. Cui,
Multigrid methods for two-dimensional Maxwell's equations on graded meshes, J. Comput. Appl. Math., 255 (2014), 231-247.
doi: 10.1016/j.cam.2013.05.007. |
[12] |
Y. Deng, H. Liu and G. Uhlmann,
Full and partial cloaking in electromagnetic scattering, Arch. Ration. Mech. Anal., 223 (2017), 265-299.
doi: 10.1007/s00205-016-1035-6. |
[13] |
W. Dörfler,
A convergent adaptive algorithm for Poisson's equation, SIAM J. Numer. Anal., 33 (1996), 1106-1124.
doi: 10.1137/0733054. |
[14] |
Y. Hao and R. Mittra, FDTD modeling of metamaterials: Theory and applications, Artech. House., (2008). Google Scholar |
[15] |
B. He, W. Yang and H. Wang, Convergence analysis of adaptive edge finite element method for variable coefficient time-harmonic Maxwell's equations, J. Comput. Appl. Math., 376 (2020), 16pp.
doi: 10.1016/j.cam.2020.112860. |
[16] |
Y. Huang, J. Li and C. Wu,
The averaging technique for superconvergence: Verification and application of 2D edge elements to Maxwell's equations in metamaterials, Comput. Methods Appl. Mech. Engrg., 255 (2013), 121-132.
doi: 10.1016/j.cma.2012.11.008. |
[17] |
Y. Huang, J. Li and W. Yang,
Interior penalty DG methods for Maxwell's equations in dispersive media, J. Comput. Phys., 230 (2011), 4559-4570.
doi: 10.1016/j.jcp.2011.02.031. |
[18] |
Y. Huang, J. Li and W. Yang, Modeling backward wave propagation in metamaterials by the finite element time-domain method, SIAM J. Sci. Comput., 35 (2013), B248–B274.
doi: 10.1137/120869869. |
[19] |
Y. Huang and N. Yi,
The superconvergent cluster recovery method, J. Sci. Comput., 44 (2010), 301-322.
doi: 10.1007/s10915-010-9379-9. |
[20] |
H. Li, J. Li and H. Liu,
On quasi-static cloaking due to anomalous localized resonance in $\mathbb{R}^{3}$, SIAM J. Appl. Math., 75 (2015), 1245-1260.
doi: 10.1137/15M1009974. |
[21] |
H. Li, S. Li, H. Liu and X. Wang,
Analysis of electromagnetic scattering from plasmonic inclusions beyond the quasi-static approximation and applications, ESAIM Math. Model. Numer. Anal., 53 (2019), 1351-1371.
doi: 10.1051/m2an/2019004. |
[22] |
J. Li and J. S. Hesthaven,
Analysis and application of the nodal discontinuous Galerkin method for wave propagation in metamaterials, J. Comput. Phys., 258 (2014), 915-930.
doi: 10.1016/j.jcp.2013.11.018. |
[23] |
J. Li, Y. Huang and W. Yang,
An adaptive edge finite element method for electromagnetic cloaking simulation, J. Comput. Phys., 249 (2013), 216-232.
doi: 10.1016/j.jcp.2013.04.026. |
[24] |
H. Liu, Virtual reshaping and invisibility in obstacle scattering, Inverse Problems, 25 (2009), 16pp.
doi: 10.1088/0266-5611/25/4/045006. |
[25] |
P. Monk, Finite Element Methods for Maxwell's Equations, Oxford University Press, New York, 2003.
doi: 10.1093/acprof:oso/9780198508885.001.0001.![]() ![]() |
[26] |
A. Naga and Z. Zhang,
A posteriori error estimates based on the polynomial preserving recovery, SIAM J. Numer. Anal., 42 (2004), 1780-1800.
doi: 10.1137/S0036142903413002. |
[27] |
N. C. Nguyen, J. Peraire and B. Cockburn,
Hybridizable discontinuous Galerkin methods for the time-harmonic Maxwell's equations, J. Comput. Phys., 230 (2011), 7151-7175.
doi: 10.1016/j.jcp.2011.05.018. |
[28] |
J. B. Pendry, D. Schurig and D. R. Smith,
Controlling electromagnetic fields, Science, 312 (2006), 1780-1782.
doi: 10.1126/science.1125907. |
[29] |
N. A. Pierce and M. B. Giles,
Adjoint recovery of superconvergent functionals from PDE approximations, SIAM. Rev., 42 (2000), 247-264.
doi: 10.1137/S0036144598349423. |
[30] |
A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, Artech House, Inc., Boston, MA, 2000. |
[31] |
H. Wang, W. Yang and Y. Huang,
Adaptive finite element method for the sound wave problems in two kinds of media, Comput. Math. Appl., 79 (2020), 789-801.
doi: 10.1016/j.camwa.2019.07.029. |
[32] |
D. H. Werner and D.-H. Kwon, Transformation Electromagnetics and Metamaterials. Fundamental Principles and Applications, Springer-Verlag, London, 2014.
doi: 10.1007/978-1-4471-4996-5. |
[33] |
W. Yang, Y. Huang and J. Li,
Developing a time-domain finite element method for the Lorentz metamaterial model and applications, J. Sci. Comput., 68 (2016), 438-463.
doi: 10.1007/s10915-015-0144-y. |
[34] |
W. Yang, J. Li and Y. Huang,
Modeling and analysis of the optical black hole in metamaterials by the finite element time-domain method, Comput. Methods Appl. Mech. Engrg., 304 (2016), 501-520.
doi: 10.1016/j.cma.2016.02.029. |
[35] |
W. Yang, J. Li and Y. Huang,
Mathematical analysis and finite element time domain simulation of arbitrary star-shaped electromagnetic cloaks, SIAM J. Numer. Anal., 56 (2018), 136-159.
doi: 10.1137/16M1093835. |
[36] |
W. Yang, J. Li, Y. Huang and B. He,
Developing finite element methods for simulating transformation optics devices with metamaterials, Commun. Comput. Phys., 25 (2019), 135-154.
doi: 10.4208/cicp.oa-2017-0225. |
[37] |
O. C. Zienkiewicz and J. Z. Zhu,
The superconvergence patch recovery (SPR) and adaptive finite element refinement, Comput. Methods. Appl. Mech. Engrg., 101 (1992), 207-224.
doi: 10.1016/0045-7825(92)90023-D. |
















Rate | Rate | |||
1/2 | 1.72241533259 | 0.65798419344 | ||
1/4 | 1.43147288047 | 0.2669 | 0.33590028941 | 0.9700 |
1/8 | 1.04238857427 | 0.4576 | 0.09973181632 | 1.7519 |
1/16 | 0.33780141151 | 1.6256 | 0.02622591011 | 1.9271 |
1/32 | 0.08957072297 | 1.9151 | 0.00685496133 | 1.9358 |
1/64 | 0.02277727214 | 1.9754 | 0.00173714021 | 1.9804 |
Rate | Rate | |||
1/2 | 1.72241533259 | 0.65798419344 | ||
1/4 | 1.43147288047 | 0.2669 | 0.33590028941 | 0.9700 |
1/8 | 1.04238857427 | 0.4576 | 0.09973181632 | 1.7519 |
1/16 | 0.33780141151 | 1.6256 | 0.02622591011 | 1.9271 |
1/32 | 0.08957072297 | 1.9151 | 0.00685496133 | 1.9358 |
1/64 | 0.02277727214 | 1.9754 | 0.00173714021 | 1.9804 |
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