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A new over-penalized weak galerkin method. Part Ⅰ: Second-order elliptic problems
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 |
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.
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. |
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