Skip to main content Accessibility help
×
Home
Hostname: page-component-7ccbd9845f-dzwm5 Total loading time: 0.396 Render date: 2023-01-28T17:45:55.398Z Has data issue: true Feature Flags: { "useRatesEcommerce": false } hasContentIssue true

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

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.

Type
Research Article
Copyright
Copyright © Global Science Press Limited 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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), 823864.3.0.CO;2-B>CrossRefGoogle 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), 17491779.CrossRefGoogle 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), 205226.CrossRefGoogle 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), 588610.CrossRefGoogle 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, 350.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.CrossRefGoogle 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), 11491176.CrossRefGoogle 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), 375386.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), 24082431.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), 440468.CrossRefGoogle 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), 257272.CrossRefGoogle 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), 186221.CrossRefGoogle Scholar
[14]Hesthaven, J. S. and Warburton, T., Nodal Discontinuous Galerkin Methods: Algorithms, Analysis and Applications, Springer, New York, 2008.CrossRefGoogle 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), 485518.CrossRefGoogle Scholar
[16]Huang, Y. and Li, J., Interior penalty discontinuous Galerkin method for Maxwell’s equation in cold plasma, J. Sci. Comput., 41 (2009), 321340.CrossRefGoogle 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), 30813094.CrossRefGoogle 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), 42204229.CrossRefGoogle 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), 12031221.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), 549580.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), 67956820.CrossRefGoogle Scholar
19
Cited by

Save article to Kindle

To save this article to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

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

Save article to Dropbox

To save this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Dropbox account. Find out more about saving content to Dropbox.

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

Save article to Google Drive

To save this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Google Drive account. Find out more about saving content to Google Drive.

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

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *