June  2020, 28(2): 961-976. doi: 10.3934/era.2020051

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

Received  February 2020 Revised  April 2020 Published  June 2020

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.

Citation: Hao Wang, Wei Yang, Yunqing Huang. An adaptive edge finite element method for the Maxwell's equations in metamaterials. Electronic Research Archive, 2020, 28 (2) : 961-976. doi: 10.3934/era.2020051
References:
[1]

K. AndoH. 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.  Google Scholar

[2]

G. BaoH. 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.  Google Scholar

[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.  Google Scholar

[4]

E. BlåstenH. LiH. 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.  Google Scholar

[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.   Google Scholar

[6]

S. C. BrennerJ. 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.  Google Scholar

[7]

S. C. BrennerJ. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

Y. DengH. 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.  Google Scholar

[13]

W. Dörfler, A convergent adaptive algorithm for Poisson's equation, SIAM J. Numer. Anal., 33 (1996), 1106-1124.  doi: 10.1137/0733054.  Google Scholar

[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.  Google Scholar

[16]

Y. HuangJ. 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.  Google Scholar

[17]

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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.  Google Scholar

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Y. Huang and N. Yi, The superconvergent cluster recovery method, J. Sci. Comput., 44 (2010), 301-322.  doi: 10.1007/s10915-010-9379-9.  Google Scholar

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H. LiJ. 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.  Google Scholar

[21]

H. LiS. LiH. 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.  Google Scholar

[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.  Google Scholar

[23]

J. LiY. 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.  Google Scholar

[24]

H. Liu, Virtual reshaping and invisibility in obstacle scattering, Inverse Problems, 25 (2009), 16pp. doi: 10.1088/0266-5611/25/4/045006.  Google Scholar

[25] P. Monk, Finite Element Methods for Maxwell's Equations, Oxford University Press, New York, 2003.  doi: 10.1093/acprof:oso/9780198508885.001.0001.  Google Scholar
[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.  Google Scholar

[27]

N. C. NguyenJ. 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.  Google Scholar

[28]

J. B. PendryD. Schurig and D. R. Smith, Controlling electromagnetic fields, Science, 312 (2006), 1780-1782.  doi: 10.1126/science.1125907.  Google Scholar

[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.  Google Scholar

[30]

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, Artech House, Inc., Boston, MA, 2000.  Google Scholar

[31]

H. WangW. 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.  Google Scholar

[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.  Google Scholar

[33]

W. YangY. 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.  Google Scholar

[34]

W. YangJ. 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.  Google Scholar

[35]

W. YangJ. 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.  Google Scholar

[36]

W. YangJ. LiY. 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.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

K. AndoH. 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.  Google Scholar

[2]

G. BaoH. 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.  Google Scholar

[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.  Google Scholar

[4]

E. BlåstenH. LiH. 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.  Google Scholar

[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.   Google Scholar

[6]

S. C. BrennerJ. 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.  Google Scholar

[7]

S. C. BrennerJ. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

Y. DengH. 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.  Google Scholar

[13]

W. Dörfler, A convergent adaptive algorithm for Poisson's equation, SIAM J. Numer. Anal., 33 (1996), 1106-1124.  doi: 10.1137/0733054.  Google Scholar

[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.  Google Scholar

[16]

Y. HuangJ. 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.  Google Scholar

[17]

Y. HuangJ. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

H. LiJ. 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.  Google Scholar

[21]

H. LiS. LiH. 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.  Google Scholar

[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.  Google Scholar

[23]

J. LiY. 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.  Google Scholar

[24]

H. Liu, Virtual reshaping and invisibility in obstacle scattering, Inverse Problems, 25 (2009), 16pp. doi: 10.1088/0266-5611/25/4/045006.  Google Scholar

[25] P. Monk, Finite Element Methods for Maxwell's Equations, Oxford University Press, New York, 2003.  doi: 10.1093/acprof:oso/9780198508885.001.0001.  Google Scholar
[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.  Google Scholar

[27]

N. C. NguyenJ. 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.  Google Scholar

[28]

J. B. PendryD. Schurig and D. R. Smith, Controlling electromagnetic fields, Science, 312 (2006), 1780-1782.  doi: 10.1126/science.1125907.  Google Scholar

[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.  Google Scholar

[30]

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, Artech House, Inc., Boston, MA, 2000.  Google Scholar

[31]

H. WangW. 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.  Google Scholar

[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.  Google Scholar

[33]

W. YangY. 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.  Google Scholar

[34]

W. YangJ. 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.  Google Scholar

[35]

W. YangJ. 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.  Google Scholar

[36]

W. YangJ. LiY. 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.  Google Scholar

[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.  Google Scholar

Figure 1.  Example 4.2: The first line and the second line are the real values of $ E_1 $ and the meshes, respectively. From left to right: $ 8510 $ Ndof (for the initial mesh), $ 133620 $ Ndof (by using $ \eta^{r0}_{K_l} $) after $ 14 $ refinements, $ 139743 $ Ndof (by using $ \eta^{r1}_{K_l} $) and $ 132334 $ Ndof (by using $ \eta^{r2}_{K_l} $) with the same times of 12 refinements
Figure 2.  Example 4.2: Snapshots of numerical solution and adaptive meshes for the real values of $ E_1 $ after $ 10 $ refinements. First two columns: $ 142833 $ Ndof (by using $ \eta^{r1}_{K_l} $); The last two columns: $ 138064 $ Ndof (by using $ \eta^{r2}_{K_l} $)
Figure 3.  Example 4.3: The first is the initial mesh with $ 6090 $ Ndof and the last three are the real values of $ E_1 $ based on the initial mesh
Figure 4.  Example 4.3: The first line and the second line are the real values of $ E_1 $ and the meshes with $ (x_0, y_0) = (1, 1.45) $ and $ m_p = 2 $, respectively. From left to right: $ 299395 $ Ndof (using $ \eta^{r0}_{K_l} $) after $ 18 $ refinements, $ 315182 $ Ndof (by using $ \eta^{r1}_{K_l} $) and $ 273473 $ Ndof (by using $ \eta^{r2}_{K_l} $) with the same times of $ 13 $ refinements
Figure 5.  Example 4.3: Snapshots of numerical solution and adaptive meshes for the real values of $ E_1 $ after $ 13 $ refinements with $ (x_0, y_0) = (1, 1.45) $ and $ m_p = 2 $. First two columns: $ 303142 $ Ndof (by using $ \eta^{r1}_{K_l} $); The last two columns: $ 278572 $ Ndof (by using $ \eta^{r2}_{K_l} $)
Figure 6.  Example 4.3: The first line and the second line are the real values of $ E_1 $ and the meshes with $ (x_0, y_0) = (-1, 1.45) $ and $ m_p = -2 $, respectively. From left to right: $ 265615 $ Ndof (by using $ \eta^{r0}_{K_l} $) after $ 21 $ refinements, $ 282153 $ Ndof (by using $ \eta^{r1}_{K_l} $) and $ 237085 $ Ndof (by using $ \eta^{r2}_{K_l} $) with the same times of $ 14 $ refinements
Figure 7.  Example 4.3: Snapshots of numerical solution and adaptive meshes for the real values of $ E_1 $ after $ 14 $ refinements with $ (x_0, y_0) = (-1, 1.45) $ and $ m_p = -2 $. First two columns: $ 282422 $ Ndof (by using $ \eta^{r1}_{K_l} $); The last two columns: $ 245356 $ Ndof (by using $ \eta^{r2}_{K_l} $)
Figure 8.  Example 4.3: The first line and the second line are the real values of $ E_1 $ and the meshes, respectively. From left to right: $ 323340 $ Ndof (by using $ \eta^{r0}_{K_l} $) after $ 21 $ refinements, $ 306934 $ Ndof (by using $ \eta^{r1}_{K_l} $) and $ 280265 $ Ndof (by using $ \eta^{r2}_{K_l} $) with the same times of $ 13 $ refinements
Figure 9.  Example 4.3: Snapshots of numerical solution and adaptive meshes for the real values of $ E_1 $ after $ 13 $ refinements. First two columns: $ 284570 $ Ndof (by using $ \eta^{r1}_{K_l} $); The last two columns: $ 271643 $ Ndof (by using $ \eta^{r2}_{K_l} $)
Figure 10.  Example 4.4: The first line, the second line and the third line are the real values of $ E_1 $, the real values of $ E_2 $ and the meshes, respectively. From left to right: $ 344405 $ Ndof (by using $ \eta^{r0}_{K_l} $) after $ 3 $ refinements, $ 344620 $ Ndof (by using $ \eta^{r1}_{K_l} $) and $ 332141 $ Ndof (by using $ \eta^{r2}_{K_l} $) with the same times of $ 17 $ refinements
Figure 11.  Example 4.4: The first column, the second column and the third column are snapshots of numerical solution for the real values of $ E_1 $, $ E_2 $ and adaptive meshes, respectively. The first line: $ 393423 $ Ndof after $ 23 $ refinements (by using $ \eta^{r1}_{K_l} $); The second line: $ 416438 $ Ndof after $ 21 $ refinements (by using $ \eta^{r2}_{K_l} $)
Figure 12.  Example 4.5: The first line and the second line are the real values of $ E_2 $ and the meshes, respectively. From left to right: $ 794533 $ Ndof (by using $ \eta^{r0}_{K_l} $) after $ 4 $ refinements, $ 506294 $ Ndof (by using $ \eta^{r1}_{K_l} $) after $ 23 $ refinements and $ 505234 $ Ndof (by using $ \eta^{r2}_{K_l} $) after $ 17 $ refinements
Figure 13.  Example 4.5: Snapshots of numerical solution and adaptive meshes for the real values of $ E_2 $. First two columns: $ 468957 $ Ndof (by using $ \eta^{r1}_{K_l} $) after $ 28 $ refinements; The last two columns: $ 656397 $ Ndof (by using $ \eta^{r2}_{K_l} $) after $ 25 $ refinements
Figure 14.  Example 4.6: The computational domain for the cloak simulation
Figure 15.  Example 4.6: The first line and the second line are the real values of $ E_2 $ and the meshes, respectively. From left to right: $ 445224 $ Ndof (by using $ \eta^{r0}_{K_l} $) after $ 126 $ refinements, $ 323420 $ Ndof (by using $ \eta^{r1}_{K_l} $) after $ 10 $ refinements, $ 120272 $ Ndof (by using $ \eta^{r2}_{K_l} $) after $ 30 $ refinements
Figure 16.  Example 4.6: Snapshots of numerical solution and adaptive meshes for the real values of $ E_2 $. First two columns: $ 291690 $ Ndof (by using $ \eta^{r1}_{K_l} $) after $ 10 $ refinements; The last two columns: $ 78497 $ Ndof (by using $ \eta^{r2}_{K_l} $) after $ 34 $ refinements
Table 1.  The Discrete $ l_2 $ errors and convergence rate
$ h $ $ ||R(\mu^{-1}\nabla\times \boldsymbol{E}) - \mu^{-1}\nabla\times \boldsymbol{E}||_{l_2} $ Rate $ ||R(\varepsilon \boldsymbol{E})-\varepsilon \boldsymbol{E}||_{l_2} $ Rate
1/2 1.72241533259 $ \ast $ 0.65798419344 $ \ast $
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
$ h $ $ ||R(\mu^{-1}\nabla\times \boldsymbol{E}) - \mu^{-1}\nabla\times \boldsymbol{E}||_{l_2} $ Rate $ ||R(\varepsilon \boldsymbol{E})-\varepsilon \boldsymbol{E}||_{l_2} $ Rate
1/2 1.72241533259 $ \ast $ 0.65798419344 $ \ast $
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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