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May  2021, 26(5): 2429-2440. doi: 10.3934/dcdsb.2020185

Convergence analysis of an accurate and efficient method for nonlinear Maxwell's equations

Jiangxing Wang 1,2,,

1. 

MOE-LCSM, School of Mathematics and Statistics, Hunan Normal University, Changsha, Hunan 410081, China
2. 

Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, 637371, Singapore

* Corresponding author: Jiangxing Wang

Received  January 2020 Revised  March 2020 Published  May 2021 Early access  June 2020

Fund Project: This work was supported in part by the NSFC No.11801171, Hunan Provincial Natural Science Foundation of China(NO. 2019JJ50384), Scientific Research Found of Hunan Provincial Education Department(No.18B023)and Singapore MOE AcRF Tier 2 Grants: MOE2017-T2-2-144 and MOE2018-T2-1-059

In this paper, we investigate an accurate and efficient method for nonlinear Maxwell's equation. DG method and Crank-Nicolson scheme are employed for spatial and time discretization, respectively. A semi-explicit extrapolation technique is adopted for the linearization of the nonlinear term. Since the proposed scheme is semi-implicit, only a linear system needs to be solved at each time step. Optimal convergent order of $ O(\tau^2+h^{p+\frac{1}{2}}) $ is proved under time step size condition $ \tau\leq h^{d/4} $. Finally, 2D and 3D numerical examples are provided to validate the theoretical convergence rate.

Keywords: Discontinuous Galerkin method, accurate and efficient, convergence analysis, fully discrete.
Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.
Citation: Jiangxing Wang. Convergence analysis of an accurate and efficient method for nonlinear Maxwell's equations. Discrete and Continuous Dynamical Systems - B, 2021, 26 (5) : 2429-2440. doi: 10.3934/dcdsb.2020185
References:
[1]

S. Brenner and R. Scott, The Mathematical Theory of Finite Element Methods, Springer Science & Business Media, 2007. doi: 10.1007/978-0-387-75934-0.

[2]

T. Chen, T. Kang, G. Lu and L. Wu, A $(T, \psi)-\psi_e$ decoupled scheme for a time-dependent multiply-connected eddy current problem, Math. Method. Appl. Sci., 37 (2014), 343-359.  doi: 10.1002/mma.2795.

[3]

B. Cockburn, F. Li and C. W.-Shu, Locally divergence-free discontinuous Galerkin methods for the Maxwell equations, J. Comp. Phys., 194 (2004), 488-610.  doi: 10.1016/j.jcp.2003.09.007.

[4]

B. Cockburn and C. W.-Shu, Runge–Kutta discontinuous Galerkin methods for convection-dominated problems, J. Sci. Comp., 16 (2001), 173-261.  doi: 10.1023/A:1012873910884.

[5]

S. Durand and M. Slodicka, Fully discrete finite element method for Maxwell's equations with nonlinear conductivity, IMA J. Numer. Anal., 31 (2011), 1713-1733.  doi: 10.1093/imanum/drr007.

[6]

S. Durand and M. Slodicka, Convergence of the mixed finite element method for Maxwell's equations with nonlinear conductivity, Math. Methods in the Applied Sciences, 35 (2012), 1489-1504.  doi: 10.1002/mma.2513.

[7]

M. Ferreira and C. Buriol, Orthogonal decomposition and asymptotic behavior for nonlinear Maxwell's equations, J. Math. Anal. Appl., 426 (2015), 392-405.  doi: 10.1016/j.jmaa.2014.12.071.

[8]

L. Fezoui, S. Lanteri, S. Lohrengel and S. Piperno, Convergence and stability of a discontinuous Galerkin time-domain method for the 3D heterogeneous Maxwell equations on unstructured meshes, ESAIM: Model. Math. Anal. Numer., 39 (2005), 1149-1176.  doi: 10.1051/m2an:2005049.

[9]

G. Gruner, A. Zawadowski and P. Chaikin, Nonlinear conductivity and noise due to charge-density-wave depinning in nb se 3, Physical Review Letters, 46 (1981), 511-515. 

[10]

R. Guo, L. Ji and Y. Xu, High order local local discontinuous Galerkin method for the Allen-Cahn equation: Analysis and simulation, J. Comput. Math., 34 (2016), 135-158.  doi: 10.4208/jcm.1510-m2014-0002.

[11]

J. Hesthaven and T. Warburton, Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications, Springer, 2008. doi: 10.1007/978-0-387-72067-8.

[12]

H. Jia, J. Li, Z. Fang and M. Li, A new FDTD scheme for Maxwell's equations in Kerr-type nonlinear media, Numerical Algorithms, 82 (2019), 223-243.  doi: 10.1007/s11075-018-0602-3.

[13]

J. Li and J. S. Hesthaven, Analysis and application of the nodal discontinuous Galerkin method for wave propagation in metamaterials, J. Comp. Phys., 258 (2014), 915-930.  doi: 10.1016/j.jcp.2013.11.018.

[14]

A. C. Metaxas and R. J. Meredith, Industrial Microwave Heating, Peter Peregrinus Ltd., London, 1983.

[15]

T. Kang, Y. Wang, L. Wu and K. Kim, An improved error estimate for Maxwell's equations with a power-law nonlinear conductivity, Appl. Math. Lett., 45 (2015), 93-97.  doi: 10.1016/j.aml.2015.01.017.

[16]

W. Reed and T. Hill, Triangular mesh methods for the neutron transport equation, Los Alamos Report LA-UR-73-479, (1973).

[17]

J. Rhyner, Magnetic properties and AC-losses of superconductors with power law current voltage characteristics, Physica C: Superconductivity, 212 (1993), 292-300.  doi: 10.1016/0921-4534(93)90592-E.

[18]

M. Slodicka, A time discretization scheme for a non-linear degenerate eddy current model for ferromagnetic materials, IMA J. Numer. Anal., 26 (2006), 173-186.  doi: 10.1093/imanum/dri030.

[19]

M. Slodicka and S. Durand, Fully discrete finite element scheme for Maxwell's equations with non-linear boundary condition, J. Math. Anal. Appl., 375 (2011), 230-244.  doi: 10.1016/j.jmaa.2010.09.016.

[20]

H. Song and C.-W. Shu, Uncodntional energy stability analysis of a second order implicit-explicit local discontinuous Galerkin method for the Cahn-Hilliard equation, J. Sci. Comput., 73 (2017), 1178-1203.  doi: 10.1007/s10915-017-0497-5.

[21]

B. Wang, Z. Xie and Z. Zhang, Error analysis of a discontinuous Galerkin method for Maxwell equations in dispersive media, J. Comput. Phys., 229 (2010), 8552-8563.  doi: 10.1016/j.jcp.2010.07.038.

[22]

B. Wang, Z. Xie and Z. Zhang, Space-time discontinuous galerkin method for maxwell equations in dispersive media, Acta Math. Sci., 34 (2014), 1357-1376.  doi: 10.1016/S0252-9602(14)60089-8.

[23]

J. Wang, Z. Xie and C. Chen, Implicit DG method for time domain Maxwell's equations involving metamaterials, Adv. Appl. Math. Mech., 7 (2015), 796-817.  doi: 10.4208/aamm.2014.m725.

[24]

Z. Xie, J. Wang, C. Chen and B. Wang, Solving Maxwell's equation in meta-materials by a CG-DG method, Commun. Comput. Phys., 19 (2016), 1242-1264.  doi: 10.4208/cicp.scpde14.35s.

[25]

C. Yao, Y. Lin, C. Wang and Y. Kou, A third order linearized BDFscheme for Maxwell's equations with nonlinear conductivity using finite element method, Int. J. Numer. Anal. Mod., 14 (2017), 511-531. 

[26]

H. Yin, On a singular limit problem for nonlinear Maxwell's equations, J. Differential Equations, 156 (1999), 355-375.  doi: 10.1006/jdeq.1998.3608.

show all references

References:
[1]

S. Brenner and R. Scott, The Mathematical Theory of Finite Element Methods, Springer Science & Business Media, 2007. doi: 10.1007/978-0-387-75934-0.

[2]

T. Chen, T. Kang, G. Lu and L. Wu, A $(T, \psi)-\psi_e$ decoupled scheme for a time-dependent multiply-connected eddy current problem, Math. Method. Appl. Sci., 37 (2014), 343-359.  doi: 10.1002/mma.2795.

[3]

B. Cockburn, F. Li and C. W.-Shu, Locally divergence-free discontinuous Galerkin methods for the Maxwell equations, J. Comp. Phys., 194 (2004), 488-610.  doi: 10.1016/j.jcp.2003.09.007.

[4]

B. Cockburn and C. W.-Shu, Runge–Kutta discontinuous Galerkin methods for convection-dominated problems, J. Sci. Comp., 16 (2001), 173-261.  doi: 10.1023/A:1012873910884.

[5]

S. Durand and M. Slodicka, Fully discrete finite element method for Maxwell's equations with nonlinear conductivity, IMA J. Numer. Anal., 31 (2011), 1713-1733.  doi: 10.1093/imanum/drr007.

[6]

S. Durand and M. Slodicka, Convergence of the mixed finite element method for Maxwell's equations with nonlinear conductivity, Math. Methods in the Applied Sciences, 35 (2012), 1489-1504.  doi: 10.1002/mma.2513.

[7]

M. Ferreira and C. Buriol, Orthogonal decomposition and asymptotic behavior for nonlinear Maxwell's equations, J. Math. Anal. Appl., 426 (2015), 392-405.  doi: 10.1016/j.jmaa.2014.12.071.

[8]

L. Fezoui, S. Lanteri, S. Lohrengel and S. Piperno, Convergence and stability of a discontinuous Galerkin time-domain method for the 3D heterogeneous Maxwell equations on unstructured meshes, ESAIM: Model. Math. Anal. Numer., 39 (2005), 1149-1176.  doi: 10.1051/m2an:2005049.

[9]

G. Gruner, A. Zawadowski and P. Chaikin, Nonlinear conductivity and noise due to charge-density-wave depinning in nb se 3, Physical Review Letters, 46 (1981), 511-515. 

[10]

R. Guo, L. Ji and Y. Xu, High order local local discontinuous Galerkin method for the Allen-Cahn equation: Analysis and simulation, J. Comput. Math., 34 (2016), 135-158.  doi: 10.4208/jcm.1510-m2014-0002.

[11]

J. Hesthaven and T. Warburton, Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications, Springer, 2008. doi: 10.1007/978-0-387-72067-8.

[12]

H. Jia, J. Li, Z. Fang and M. Li, A new FDTD scheme for Maxwell's equations in Kerr-type nonlinear media, Numerical Algorithms, 82 (2019), 223-243.  doi: 10.1007/s11075-018-0602-3.

[13]

J. Li and J. S. Hesthaven, Analysis and application of the nodal discontinuous Galerkin method for wave propagation in metamaterials, J. Comp. Phys., 258 (2014), 915-930.  doi: 10.1016/j.jcp.2013.11.018.

[14]

A. C. Metaxas and R. J. Meredith, Industrial Microwave Heating, Peter Peregrinus Ltd., London, 1983.

[15]

T. Kang, Y. Wang, L. Wu and K. Kim, An improved error estimate for Maxwell's equations with a power-law nonlinear conductivity, Appl. Math. Lett., 45 (2015), 93-97.  doi: 10.1016/j.aml.2015.01.017.

[16]

W. Reed and T. Hill, Triangular mesh methods for the neutron transport equation, Los Alamos Report LA-UR-73-479, (1973).

[17]

J. Rhyner, Magnetic properties and AC-losses of superconductors with power law current voltage characteristics, Physica C: Superconductivity, 212 (1993), 292-300.  doi: 10.1016/0921-4534(93)90592-E.

[18]

M. Slodicka, A time discretization scheme for a non-linear degenerate eddy current model for ferromagnetic materials, IMA J. Numer. Anal., 26 (2006), 173-186.  doi: 10.1093/imanum/dri030.

[19]

M. Slodicka and S. Durand, Fully discrete finite element scheme for Maxwell's equations with non-linear boundary condition, J. Math. Anal. Appl., 375 (2011), 230-244.  doi: 10.1016/j.jmaa.2010.09.016.

[20]

H. Song and C.-W. Shu, Uncodntional energy stability analysis of a second order implicit-explicit local discontinuous Galerkin method for the Cahn-Hilliard equation, J. Sci. Comput., 73 (2017), 1178-1203.  doi: 10.1007/s10915-017-0497-5.

[21]

B. Wang, Z. Xie and Z. Zhang, Error analysis of a discontinuous Galerkin method for Maxwell equations in dispersive media, J. Comput. Phys., 229 (2010), 8552-8563.  doi: 10.1016/j.jcp.2010.07.038.

[22]

B. Wang, Z. Xie and Z. Zhang, Space-time discontinuous galerkin method for maxwell equations in dispersive media, Acta Math. Sci., 34 (2014), 1357-1376.  doi: 10.1016/S0252-9602(14)60089-8.

[23]

J. Wang, Z. Xie and C. Chen, Implicit DG method for time domain Maxwell's equations involving metamaterials, Adv. Appl. Math. Mech., 7 (2015), 796-817.  doi: 10.4208/aamm.2014.m725.

[24]

Z. Xie, J. Wang, C. Chen and B. Wang, Solving Maxwell's equation in meta-materials by a CG-DG method, Commun. Comput. Phys., 19 (2016), 1242-1264.  doi: 10.4208/cicp.scpde14.35s.

[25]

C. Yao, Y. Lin, C. Wang and Y. Kou, A third order linearized BDFscheme for Maxwell's equations with nonlinear conductivity using finite element method, Int. J. Numer. Anal. Mod., 14 (2017), 511-531. 

[26]

H. Yin, On a singular limit problem for nonlinear Maxwell's equations, J. Differential Equations, 156 (1999), 355-375.  doi: 10.1006/jdeq.1998.3608.

Figure 1.  $ k = 1, \tau = h $ (Left), $ k = 2, \tau = h^2 $ (Right)
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Figure 2.  $ p = 1, \tau = h $ (Left), $ p = 2, \tau = h^2 $ (Right)
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