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High-Order Low Dissipation Conforming Finite-Element Discretization of the Maxwell Equations

Published online by Cambridge University Press:  20 August 2015

Sébastien Jund ,
Stéphanie Salmon  and
Eric Sonnendrücker
Sébastien Jund
Affiliation:
IRMA, UMR 7501 Université de Strasbourg and CNRS, and CALVI project-team, INRIA Nancy Grand Est, 7 rue René Descartes, F-67084 STRASBOURG Cedex
Stéphanie Salmon*
Affiliation:
Laboratoire de Mathématiques EA 4535, Université de Reims, U.F.R. Sciences Exactes et Naturelles, Moulin de la Housse – BP 1039, F-51687 REIMS cedex 2
Eric Sonnendrücker*
Affiliation:
IRMA, UMR 7501 Université de Strasbourg and CNRS, and CALVI project-team, INRIA Nancy Grand Est, 7 rue René Descartes, F-67084 STRASBOURG Cedex
*
Email:stephanie.salmon@univ-reims.fr
Corresponding author.Email:sonnen@math.unistra.fr
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Abstract

In this paper, we study high order discretization methods for solving the Maxwell equations on hybrid triangle-quad meshes. We have developed high order finite edge element methods coupled with different high order time schemes and we compare results and efficiency for several schemes. We introduce in particular a class of simple high order low dissipation time schemes based on a modified Taylor expansion.

Keywords

65M60Maxwell’s equationsedge finite element methodmass lumpingtime discretization schemes
Type
Research Article
Information
Communications in Computational Physics , Volume 11 , Issue 3 , March 2012 , pp. 863 - 892
Copyright
Copyright © Global Science Press Limited 2012

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References

[1]Assous, F., Degond, P., Heintze, E., Raviart, P.-A., Segré, J., On a finite-element method for solving the three-dimensional Maxwell equations, J. Comput. Phys. 109 (1993), no. 2, 222–237.Google Scholar
[2]Canouet, N., Fezoui, L., Piperno, S., Discontinuous Galerkin time-domain solution of Maxwell’s equations on locally-refined nonconforming Cartesian grids, COMPEL 24 (2005), no. 4, 1381–1401.Google Scholar
[3]Castillo, P., Rieben, R., White, D., FEMSTER: An object oriented class library of high-order discrete differential forms, ACM Trans. Math. Softw. 31 (2005), no. 4.Google Scholar
[4]Channell, P. J., Scovel, C., Symplectic integration of Hamiltonian systems, Nonlinearity 3 (1990), 231–259.CrossRefGoogle Scholar
[5]Chin-Joe-Kong, M. J. S., Mulder, W. A., Van Veldhuizen, M., Higher-order triangular and tetra-hedral finite elements with mass lumping for solving the wave equation, J. Engrg. Math. 35 (1999), no 4, 405–426.Google Scholar
[6]Cohen, G., Higher-Order Numerical Methods for Transient Wave equation, Springer-Verlag, 2001.Google Scholar
[7]Cohen, G., Duruflé, M., Non spurious spectral-like element methods for Maxwell’s equations, J. Comput. Math. 25 (2007), no. 3, 282–304.Google Scholar
[8]Cohen, G., Monk, P., Mur-Nédélec finite element schemes for Maxwell’s equations, Comput. Methods Appl. Mech. Engrg. 169 (1999), no. 3-4, 197–217.Google Scholar
[9]Cohen, G., Monk, P., Gauss point mass lumping schemes for Maxwell’s equations, Numer. Meth. Part. D. E. J. 14 (1998), No. 1, 63–88.Google Scholar
[10]Dablain, M. A., The application of high order differencing for the scalar wave equation, Geophysics 51 (1986), No. 1, 54–66.Google Scholar
[11]Gilbert, J.C., Joly, P., Higher order time stepping for second order hyperbolic problems and optimal CFL conditions, Partial differential equations, 67–93, Comput. Methods Appl. Sci. 16, Springer, Dordrecht, 2008.Google Scholar
[12]Dumbser, M., Munz, C.-D., Arbitrary High Order Discontinuous Galerkin Schemes, Numerical Methods for Hyperbolic and Kinetic Problems, eds. Cordier, S., Goudon, T., Gutnic, M., Sonnendrücker, E., IRMA series in Mathematics and Theoretical Physics, European Mathematical Society, 2005.Google Scholar
[13]Dumbser, M., Munz, C.-D., ADER discontinuous Galerkin schemes for aeroacoustics, C. R. Mécanique 333-9 (2005), 683–687.CrossRefGoogle Scholar
[14]Elmkies, A., Joly, P., Edge finite elements and mass lumping for Maxwell’s equations: The 2D case, C. R. Acad. Sci. Paris 324 (1997), Série I, 1287–1293.CrossRefGoogle Scholar
[15]Elmkies, A., Joly, P., Edge finite elements and mass lumping for Maxwell’s equations: The 3D case, C. R. Acad. Sci. Paris 325 (1997), Série I, 1217–1222.Google Scholar
[16]Fezoui, L., Lanteri, S., Lohrengel, S., Piperno, S., Convergence and stability of a discontinuous Galerkin time-domain method for the 3D heterogeneous Maxwell equations on unstructured meshes, Math. Model. Numer. Anal. 39 (2005), no. 6, 1149–1176.CrossRefGoogle Scholar
[17]Fisher, A., Rieben, R. N., Rodrigue, G., White, D. A., A generalized mass lumping technique for vector finite-element solutions of the time-dependent Maxwell equations, IEEE T. Antenn. Propag. 53 (2005), no. 9, 2900–2910.Google Scholar
[18]Gaitonde, D., Shang, J. S., Optimized compact-difference-based finite-volume schemes for linear wave phenomena, J. Comput. Phys. 138 (1997), no. 2, 617–643.CrossRefGoogle Scholar
[19]Gottlieb, S., Gottlieb, L.-A. J., Strong stability preserving properties of Runge-Kutta time discretization methods for linear constant coefficient operators, J. Sci. Comput. 18 (2003), no. 1, 83–109.CrossRefGoogle Scholar
[20]Gottlieb, S., Shu, C.-W., Tadmor, E., Strong stability-preserving high-order time discretization methods, SIAM Rev. 43 (2001), no. 1, 89–112.Google Scholar
[21]Harten, A., Engquist, B., Osher, S., Chakravarthy, S., Uniformly high order essentially non-oscillatory schemes, III, J. Comput. Phys. 71 (1987), 231–303.CrossRefGoogle Scholar
[22]Hesthaven, J. S., High-Order Accurate Methods in Time-Domain Computational Electromagnetics. A Review, Adv. Imag. Electron Phys. 127 (2003), 59–123.Google Scholar
[23]Hesthaven, J. S. and Warburton, T., Nodal high-order methods on unstructured grids, I. Timedomain solution of Maxwell’s equations, J. Comput. Phys. 181 (2002), 186–221.CrossRefGoogle Scholar
[24]Hesthaven, J. S. and Warburton, T., Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications, Springer (2007).Google Scholar
[25]Hiptmair, R., Finite elements in computational electromagnetism, Acta Numerica (2002), 237–339.Google Scholar
[26]Hu, F. Q., Hussaini, M. Y., Manthey, J. L., Low-dissipation and low-dispersion Runge-Kutta schemes for computational acoustics, J. Comput. Phys. 124 (1996), 177–191.Google Scholar
[27]Lacoste, P., Mass-lumping for first-order Raviart-Thomas-Nédélec mixed finite elements, C. R. Math. Acad. Sci. Paris 339 (2004), no. 10, 727–732.Google Scholar
[28]Monk, P., Finite Element Methods for Maxwell’s Equations, Oxford University Press, 2003.CrossRefGoogle Scholar
[29]Nédélec, J.-C., Mixed finite elements in R3, Numer. Math. 35 (1980), 315–341.Google Scholar
[30]Pernet, S., Ferrieres, X., Cohen, G., High spatial order finite element method to solve Maxwell’s equations in time domain, IEEE T. Antenn. Propag. 53 (2005), no. 9, 2889–2899.Google Scholar
[31]Piperno, S., Symplectic local time-stepping in non-dissipative DGTD methods applied to wave propagation problems, Math. Model. Numer. Anal. 40 (2006), no. 5, 815–841.Google Scholar
[32]Rieben, R., White, D., and Rodrigue, G., High Order Symplectic Integration Methods for Finite Element Solutions to Time Dependent Maxwell Equations, IEEE T. Antenn. Propag. 52 (2004), no. 8, 2190–2195.CrossRefGoogle Scholar
[33]Rodrigue, G., White, D., A vector finite element time-domain method for solving Maxwell’s equations on unstructured hexaedral grids, SIAM J. Sci. Comput. 23 (2001) no. 3, 683–706.Google Scholar
[34]Shubin, G. R., Bell, J. B., A modified equation approach to constructing fourth-order methods for acoustic wave propagation, SIAM J. Sci. Statist. Comput. 8 (1987), no. 2, 135–151.Google Scholar
[35]Sármány, D., Botchev, M.A., van der Vegt, J. J. W., Dispersion and dissipation error in highorder Runge-Kutta discontinuous Galerkin discretizations of the Maxwell equations, J. Sci. Comput. 33 (2007), no. 1, 47–74.Google Scholar
[36]Warburton, T., An explicit construction for interpolation nodes on the simplex, J. Engrg. Math. 56 (2006), no. 3, 247–262.Google Scholar
[37]Yee, K. S., Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media, IEEE T. Antenn. Propag. 14 (1966).Google Scholar
[38]Yoshida, H., Construction of higher order symplectic integrators, Phys. Lett. A 150 (1990), no. 5-7, 262–268.Google Scholar