    ## 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:
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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  W. Reed and T. Hill, Triangular mesh methods for the neutron transport equation, Los Alamos Report LA-UR-73-479, (1973). Google Scholar  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  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  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  H. Song and C.-W. 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Phys., 19 (2016), 1242-1264.  doi: 10.4208/cicp.scpde14.35s.  Google Scholar  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  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

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##### References:
  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  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  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  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  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  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  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  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  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  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  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  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  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  A. C. Metaxas and R. J. Meredith, Industrial Microwave Heating, Peter Peregrinus Ltd., London, 1983. Google Scholar  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  W. Reed and T. Hill, Triangular mesh methods for the neutron transport equation, Los Alamos Report LA-UR-73-479, (1973). Google Scholar  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  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  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  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  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  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  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  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  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  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
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