# American Institute of Mathematical Sciences

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

 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  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.

Citation: Jiangxing Wang. Convergence analysis of an accurate and efficient method for nonlinear Maxwell's equations. Discrete & Continuous Dynamical Systems - B, 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [11] J. Hesthaven and T. Warburton, Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications, Springer, 2008. doi: 10.1007/978-0-387-72067-8.  Google Scholar [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.  Google Scholar [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.  Google Scholar [14] A. C. Metaxas and R. J. Meredith, Industrial Microwave Heating, Peter Peregrinus Ltd., London, 1983. Google Scholar [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.  Google Scholar [16] W. Reed and T. Hill, Triangular mesh methods for the neutron transport equation, Los Alamos Report LA-UR-73-479, (1973). Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar

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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [11] J. Hesthaven and T. Warburton, Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications, Springer, 2008. doi: 10.1007/978-0-387-72067-8.  Google Scholar [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.  Google Scholar [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.  Google Scholar [14] A. C. Metaxas and R. J. Meredith, Industrial Microwave Heating, Peter Peregrinus Ltd., London, 1983. Google Scholar [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.  Google Scholar [16] W. Reed and T. Hill, Triangular mesh methods for the neutron transport equation, Los Alamos Report LA-UR-73-479, (1973). Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar
$k = 1, \tau = h$ (Left), $k = 2, \tau = h^2$ (Right)
$p = 1, \tau = h$ (Left), $p = 2, \tau = h^2$ (Right)
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