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Optimal L2 Error Estimates for the Interior Penalty DG Method for Maxwell’s Equations in Cold Plasma

Published online by Cambridge University Press:  20 August 2015

Jichun Li
Jichun Li*
Affiliation:
Department of Mathematical Sciences, University of Nevada, Las Vegas, Nevada 89154-4020, USA
*
*Corresponding author.Email:jichun@unlv.nevada.edu
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Abstract

In this paper, we consider an interior penalty discontinuous Galerkin (DG) method for the time-dependent Maxwell’s equations in cold plasma. In Huang and Li (J. Sci. Comput., 42 (2009), 321-340), for both semi and fully discrete DG schemes, we proved error estimates which are optimal in the energy norm, but sub-optimal in the L2-norm. Here by filling this gap, we show that these schemes are optimally convergent in the L2-norm on quasi-uniform tetrahedral meshes if the solution is sufficiently smooth.

Keywords

65N3035L1578-08Maxwell’s equationscold plasmadiscontinuous Galerkin method

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Type
Research Article
Information
Communications in Computational Physics , Volume 11 , Issue 2 , February 2012 , pp. 319 - 334
Copyright
Copyright © Global Science Press Limited 2012

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References

[1]Amrouche, C., Bernardi, C., Dauge, M. and Girault, V., Vector potentials in three-dimensional non-smooth domains, Math. Methods Appl. Sci., 21 (1998), 823–864.Google Scholar
[2]Arnold, D. N., Brezzi, F., Cockburn, B. and Marini, L. D., Unified analysis of discontinuous Galerkin methods for elliptic problems, SIAM J. Numer. Anal., 39 (2002), 1749–1779.Google Scholar
[3]Barucq, H., Bonnet-Bendhia, A.-S., Cohen, G., Diaz, J., Ezziani, A. and Joly, P. (eds.), Proceedings of Waves 2009: The 9th International Conferences on Mathematical and Numerical Aspects of Waves Propagation, June 15-19, 2009, Pau, France.Google Scholar
[4]Chen, M.-H., Cockburn, B. and Reitich, F., High-order RKDG methods for computational electromagnetics, J. Sci. Comput., 22 (2005), 205–226.Google Scholar
[5]Cockburn, B., Li, F. and Shu, C.-W., Locally divergence-free discontinuous Galerkin methods for the Maxwell equations, J. Comput. Phys., 194 (2004), 588–610.Google Scholar
[6]Cockburn, B., Karniadakis, G. E. and Shu, C.-W., The development of discontinuous Galerkin methods, in: Discontinuous Galerkin Methods: Theory, Computation and Applications (eds. by Cockburn, B., Karniadakis, G. E. and Shu, C.-W.), Springer, Berlin, 2000, 3–50.CrossRefGoogle Scholar
[7]Demkowicz, L., Computing with hp-Adaptive Finite Elements I: One and Two-Dimensional Elliptic and Maxwell Problems, CRC Press, Taylor and Francis, 2006.Google Scholar
[8]Fezoui, L., Lanteri, S., Lohrengel, S. and Piperno, S., Convergence and stability of a discontinuous Galerkin time-domain methods for the 3D heterogeneous Maxwell equations on unstructured meshes, Model. Math. Anal. Numer., 39(6) (2005), 1149–1176.Google Scholar
[9]Grote, M. J., Schneebeli, A. and Schötzau, D., Interior penalty discontinuous Galerkin method for Maxwell’s equations: energy norm error estimates, J. Comput. Appl. Math., 204 (2007), 375–386.CrossRefGoogle Scholar
[10]Grote, M. J., Schneebeli, A. and Schötzau, D., Discontinuous Galerkin finite element method for the wave equation, SIAM J. Numer. Anal., 44 (2006), 2408–2431.CrossRefGoogle Scholar
[11]Grote, M. J., Schneebeli, A. and Schötzau, D., Interior penalty discontinuous Galerkin method for Maxwell’s equations: optimal L 2-norm error estimates, IMA J. Numer. Anal., 28 (2008), 440–468.Google Scholar
[12]Grote, M. J. and Schötzau, D., Optimal error estimates for the fully discrete interior penalty DG method for the wave equation, J. Sci. Comput., 40 (2009), 257–272.Google Scholar
[13]Hesthaven, J. S. and Warburton, T., High-order nodal methods on unstructured grids I: timedomain solution of Maxwell’s equations, J. Comput. Phys., 181 (2002), 186–221.CrossRefGoogle Scholar
[14]Hesthaven, J. S. and Warburton, T., Nodal Discontinuous Galerkin Methods: Algorithms, Analysis and Applications, Springer, New York, 2008.Google Scholar
[15]Houston, P., Perugia, I., Schneebeli, A. and Schötzau, D., Interior penalty method for the indefinite time-harmonic Maxwell equations, Numer. Math., 100 (2005), 485–518.Google Scholar
[16]Huang, Y. and Li, J., Interior penalty discontinuous Galerkin method for Maxwell’s equation in cold plasma, J. Sci. Comput., 41 (2009), 321–340.Google Scholar
[17]Li, J., Error analysisof fully discrete mixedfinite element scheme for 3-D Maxwell’s equations in dispersive media, Comput. Methods Appl. Mech. Eng., 196 (2007), 3081–3094.Google Scholar
[18]Li, J. and Chen, Y., Analysis of a time-domain finite element method for 3-D Maxwell’s equations in dispersive media, Comput. Methods Appl. Mech. Eng., 195 (2006), 4220–4229.Google Scholar
[19]Li, J. and Chen, Y., Finite element study of time-dependent Maxwell’s equations in dispersive media, Numer. Meth. Part. D. E., 24 (2008), 1203–1221.CrossRefGoogle Scholar
[20]Lu, T., Zhang, P. and Cai, W., Discontinuous Galerkin methods for dispersive and lossy Maxwell’s equations and PML boundary conditions, J. Comput. Phys., 200 (2004), 549–580.CrossRefGoogle Scholar
[21]Monk, P., Finite Element Methods for Maxwell’s Equations, Oxford University Press, 2003.Google Scholar
[22]Montseny, E., Pernet, S., Ferriéres, X. and Cohen, G., Dissipative terms and local time-stepping improvements in a spatial high order Discontinuous Galerkin scheme for the time-domain Maxwell’s equations, J. Comput. Phys., 227 (2008), 6795–6820.Google Scholar