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Stability of a Leap-Frog Discontinuous Galerkin Method for Time-Domain Maxwell's Equations in Anisotropic Materials

Part of: Partial differential equations, initial value and time-dependent initial-boundary value problems Equations of mathematical physics and other areas of application

Published online by Cambridge University Press:  27 March 2017

Adérito Araújo ,
Sílvia Barbeiro  and
Maryam Khaksar Ghalati
Adérito Araújo*
Affiliation:
CMUC, Department of Mathematics, University of Coimbra, Apartado 3008, EC Santa Cruz, 3001 – 501 Coimbra, Portugal
Sílvia Barbeiro*
Affiliation:
CMUC, Department of Mathematics, University of Coimbra, Apartado 3008, EC Santa Cruz, 3001 – 501 Coimbra, Portugal
Maryam Khaksar Ghalati*
Affiliation:
CMUC, Department of Mathematics, University of Coimbra, Apartado 3008, EC Santa Cruz, 3001 – 501 Coimbra, Portugal
*
*Corresponding author. Email addresses:alma@mat.uc.pt (A. Araújo), silvia@mat.uc.pt (S. Barbeiro), maryam@mat.uc.pt (M. Kh. Ghalati)
*Corresponding author. Email addresses:alma@mat.uc.pt (A. Araújo), silvia@mat.uc.pt (S. Barbeiro), maryam@mat.uc.pt (M. Kh. Ghalati)
*Corresponding author. Email addresses:alma@mat.uc.pt (A. Araújo), silvia@mat.uc.pt (S. Barbeiro), maryam@mat.uc.pt (M. Kh. Ghalati)
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Abstract

In this work we discuss the numerical discretization of the time-dependent Maxwell's equations using a fully explicit leap-frog type discontinuous Galerkin method. We present a sufficient condition for the stability and error estimates, for cases of typical boundary conditions, either perfect electric, perfect magnetic or first order Silver-Müller. The bounds of the stability region point out the influence of not only the mesh size but also the dependence on the choice of the numerical flux and the degree of the polynomials used in the construction of the finite element space, making possible to balance accuracy and computational efficiency. In the model we consider heterogeneous anisotropic permittivity tensors which arise naturally in many applications of interest. Numerical results supporting the analysis are provided.

Keywords

Maxwell's equationsfully explicit leap-frog discontinuous Galerkin methodstability

MSC classification

Secondary: 35Q61: Maxwell equations 65M60: Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods
Type
Research Article
Information
Communications in Computational Physics , Volume 21 , Issue 5 , May 2017 , pp. 1350 - 1375
Copyright
Copyright © Global-Science Press 2017 

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References

[1] Álvarez González, J.. A Discontinuous Galerkin Finite Element Method for the Time-Domain Solution of Maxwell Equations. PhD thesis, Universidad de Granada, 2014.Google Scholar
[2] Araújo, A., Barbeiro, S., Pinto, L., Caramelo, F., Correia, A. L., Morgado, M., Serranho, P., Silva, A. S. C., and Bernardes, R.. Maxwell's equations to model electromagnetic wave's propagation through eye's structures. Proceedings of the 13th International Conference on Computational and MathematicalMethods in Science and Engineering, CMMSE 2013, Almeria, Hamilton, Ian and Vigo-Aguiar, Jesús Eds, 1:121–129, June 2013.Google Scholar
[3] Babuška, L. and Suri, M.. The optimal convergence rate of the p-version of the finite element method. SIAM Journal on Numerical Analysis, 24(4):750–776, 1987.Google Scholar
[4] Born, M. and Wolf, E.. Principles of Optics. Cambridge University Press, 7 edition, 1999.Google Scholar
[5] Ciarlet, P. G.. The Finite Element Method for Elliptic Problems. Studies in mathematics and its applications. North-Holland, Amsterdam, New-York, 1980.Google Scholar
[6] Dobson, D. C.. An efficient method for band structure calculations in 2D photonic crystals. Journal of Computational Physics, 149(2):363–376, 1999.Google Scholar
[7] Emmrich, E.. Stability and error of the variable two-step BDF for semilinear parabolic problems. Journal of Applied Mathematics and Computing, 19(1-2):33–55, 2005.CrossRefGoogle Scholar
[8] Evans, L. C.. Partial differential equations, volume 19 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, second edition, 2010.Google Scholar
[9] Fezoui, L., Lanteri, S., Lohrengel, S., and Piperno, S.. Convergence and stability of a discontinuous Galerkin time-domain method for the 3D heterogeneous Maxwell's equations on unstructured meshes. ESAIM: Mathematical Modeling and Numerical Analysis, 39(6):1149–1176, 2005.Google Scholar
[10] Georgoulis, E.H.. Inverse-type estimates on hp-finite element spaces and applications. Mathematics of Computation, 77(261):201–219, 2008.Google Scholar
[11] Hesthaven, J. S. and Warburton, T.. Discontinuous Galerkin methods for the time-domain Maxwell's equations: An introduction. ACES Newsletter, 19:10–29, 2004.Google Scholar
[12] Hesthaven, J.S. and Warburton, T.. Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications. Springer Publishing Company, Incorporated, 1st edition, 2008.Google Scholar
[13] König, M., Busch, K., and Niegemann, J.. The discontinuous Galerkin time-domain method for Maxwell's equations with anisotropic materials. Photonics and Nanostructures-Fundamentals and Applications, 8(4):303–309, 2010.Google Scholar
[14] Leonhardt, U. and Tomáš, T.. Broadband invisibility by non-euclidean cloaking. Science, 323(5910):110–112, 2009.Google Scholar
[15] Li, J., Waters, J.W., and Machorro, E. A.. An implicit leap-frog discontinuous Galerkin method for the time-domain Maxwell's equations in metamaterials. Computer Methods in Applied Mechanics and Engineering, 223:43–54, 2012.Google Scholar
[16] Lu, T., Zhang, P., and Cai, W.. Discontinuous Galerkin methods for dispersive and lossy Maxwell's equations and PML boundary conditions. Journal of Computational Physics, 200(2):549–580, 2004.Google Scholar
[17] Mikhailov, S. A. and Ziegler, K.. New electromagnetic mode in graphene. Physical Review Letters, 99:016803, Jul 2007.Google Scholar
[18] Riviére, B.. Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations: Theory and Implementation. Society for Industrial and Applied Mathematics, Philadelphia, PA, USA, 2008.Google Scholar
[19] Riviére, B., Wheeler, M. F., and Girault, V.. A priori error estimates for finite element methods based on discontinuous approximation spaces for elliptic problems. SIAM Journal on Numerical Analysis, 39(3):902–931, 2001.Google Scholar
[20] Santos, M., Araújo, A., Barbeiro, S., Caramelo, F., Correia, A., Marques, M. I., Pinto, L., Serranho, P., Bernardes, R., and Morgado, M.. Simulation of cellular changes on optical coherence tomography of human retina. In 2015 37th Annual International Conference of the IEEE Engineering in Medicine and Biology Society (EMBC), pages 8147–8150, Aug 2015.Google Scholar
[21] Taflove, A. and Hagness, S. C.. Computational Electrodynamics: The Finite-difference Time-domain Method. Artech House antennas and propagation library. Artech House, 2005.Google Scholar
[22] Wloka, J.. Partial Differential Equations. Cambridge University Press, Cambridge, 1987.Google Scholar
[23] Yee, K. S.. Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic media. IEEE Transaction on Antennas and Propagation, 14(3):302–307, 1966.Google Scholar
[24] Yeh, P. and Gu, C.. Optics of Liquid Crystal Displays. Wiley Publishing, 2nd edition, 2009.Google Scholar