HP a-priori error estimates for a non-dissipative spectral discontinuous Galerkin method to solve the Maxwell equations in the time domain

Pernet, S.; Ferrieres, X.

doi:10.1090/S0025-5718-07-01991-6

HP a-priori error estimates for a non-dissipative spectral discontinuous Galerkin method to solve the Maxwell equations in the time domain
HTML articles powered by AMS MathViewer

by S. Pernet and X. Ferrieres PDF

Math. Comp. 76 (2007), 1801-1832 Request permission

Abstract:

In this paper, we present the $hp$-convergence analysis of a non-dissipative high-order discontinuous Galerkin method on unstructured hexahedral meshes using a mass-lumping technique to solve the time-dependent Maxwell equations. In particular, we underline the spectral convergence of the method (in the sense that when the solutions and the data are very smooth, the discretization is of unlimited order). Moreover, we see that the choice of a non-standard approximate space (for a discontinuous formulation) with the absence of dissipation can imply a loss of spatial convergence. Finally we present a numerical result which seems to confirm this property.

References

Philippe G. Ciarlet, The finite element method for elliptic problems, Studies in Mathematics and its Applications, Vol. 4, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978. MR 0520174
Achdou Yves, The Finite Element Methods, www.ann.jussieu.fr/achdou/enseignement.
Malika Remaki, Méthodes numériques pour les équations de Maxwell instationnnaires en Milieu hétérogène, Doctorat de Mathématiques Appliquées de l’Ecole Nationale des Ponts et Chaussées, 1999.
Gary C. Cohen, Higher-order numerical methods for transient wave equations, Scientific Computation, Springer-Verlag, Berlin, 2002. With a foreword by R. Glowinski. MR 1870851, DOI 10.1007/978-3-662-04823-8
K.S. Yee, Numerical solution of initial boundary value problems involving Maxwell’s equation in isotropic media. IEEE Trans. Antennas Prop., 14, 302–307, 1966.
Allen Taflove (ed.), Advances in computational electrodynamics, Artech House Antenna Library, Artech House, Inc., Boston, MA, 1998. The finite-difference time-domain method. MR 1639352
Andreas C. Cangellaris and Diana B. Wright, Analysis of the Numerical Error Caused by the Stair-Stepped Approximation of a Conducting Boundary in FDTD Simulations of Electromagnetic Phenomena, IEEE Trans. Antennas Prop., vol. AP-39, No. 10, pp. 1518-1525, October 1991.
J. S. Hesthaven and T. Warburton, Nodal high-order methods on unstructured grids. I. Time-domain solution of Maxwell’s equations, J. Comput. Phys. 181 (2002), no. 1, 186–221. MR 1925981, DOI 10.1006/jcph.2002.7118
Garry Rodrigue and Daniel White, A vector finite element time-domain method for solving Maxwell’s equations on unstructured hexahedral grids, SIAM J. Sci. Comput. 23 (2001), no. 3, 683–706. MR 1860960, DOI 10.1137/S1064827598343826
P. Bonnet and X. Ferrieres, Numerical modeling of scattering problems using a time domain finite volume method, JEWA, vol. 11, pp. 1165-1189, 1997.
Serge Piperno, Malika Remaki, and Loula Fezoui, A nondiffusive finite volume scheme for the three-dimensional Maxwell’s equations on unstructured meshes, SIAM J. Numer. Anal. 39 (2002), no. 6, 2089–2108. MR 1897951, DOI 10.1137/S0036142901387683
Bernardo Cockburn, Fengyan Li, and Chi-Wang Shu, Locally divergence-free discontinuous Galerkin methods for the Maxwell equations, J. Comput. Phys. 194 (2004), no. 2, 588–610. MR 2034859, DOI 10.1016/j.jcp.2003.09.007
Bernardo Cockburn and Chi-Wang Shu, The Runge-Kutta discontinuous Galerkin method for conservation laws. V. Multidimensional systems, J. Comput. Phys. 141 (1998), no. 2, 199–224. MR 1619652, DOI 10.1006/jcph.1998.5892
Bernardo Cockburn, George E. Karniadakis, and Chi-Wang Shu, The development of discontinuous Galerkin methods, Discontinuous Galerkin methods (Newport, RI, 1999) Lect. Notes Comput. Sci. Eng., vol. 11, Springer, Berlin, 2000, pp. 3–50. MR 1842161, DOI 10.1007/978-3-642-59721-3_{1}
Nicolas Canouet, Méthodes de Galerkin Discontinu pour la résolution du système de Maxwell sur des maillages localement raffinés non-conforme,Doctorat de Mathématiques Appliquées de l’Ecole Nationale des Ponts et Chaussées, December 2003.
Paul Houston, Ilaria Perugia, and Dominik Schötzau, Mixed discontinuous Galerkin approximation of the Maxwell operator, SIAM J. Numer. Anal. 42 (2004), no. 1, 434–459. MR 2051073, DOI 10.1137/S003614290241790X
I. Perugia, D. Schötzau, and P. Monk, Stabilized interior penalty methods for the time-harmonic Maxwell equations, Comput. Methods Appl. Mech. Engrg. 191 (2002), no. 41-42, 4675–4697. MR 1929626, DOI 10.1016/S0045-7825(02)00399-7
W. Reed and T. Hill, Triangular mesh methods for the neutron transport equation, Tech. Report LA-UR-73-479, Los Alamos National Laboratory, Los Alamos, New Mexico, USA, 1973.
P. Lasaint and P.-A. Raviart, On a finite element method for solving the neutron transport equation, Mathematical aspects of finite elements in partial differential equations (Proc. Sympos., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1974) Publication No. 33, Math. Res. Center, Univ. of Wisconsin-Madison, Academic Press, New York, 1974, pp. 89–123. MR 0658142
C. Johnson and J. Pitkäranta, An analysis of the discontinuous Galerkin method for a scalar hyperbolic equation, Math. Comp. 46 (1986), no. 173, 1–26. MR 815828, DOI 10.1090/S0025-5718-1986-0815828-4
J.-C. Nédélec, Mixed finite elements in $\textbf {R}^{3}$, Numer. Math. 35 (1980), no. 3, 315–341. MR 592160, DOI 10.1007/BF01396415
J.-C. Nédélec, A new family of mixed finite elements in $\textbf {R}^3$, Numer. Math. 50 (1986), no. 1, 57–81. MR 864305, DOI 10.1007/BF01389668
Gary Cohen and Peter Monk, Mur-Nédélec finite element schemes for Maxwell’s equations, Comput. Methods Appl. Mech. Engrg. 169 (1999), no. 3-4, 197–217. MR 1675684, DOI 10.1016/S0045-7825(98)00154-6
Peter Monk and Gerard R. Richter, A discontinuous Galerkin method for linear symmetric hyperbolic systems in inhomogeneous media, J. Sci. Comput. 22/23 (2005), 443–477. MR 2142205, DOI 10.1007/s10915-004-4132-5
S. M. Rao, Time domain electromagnetics, Series Editor, David Irwin, Auburn University, Academic Press, 1999.
Jianming Jin, The finite element method in electromagnetics, 2nd ed., Wiley-Interscience [John Wiley & Sons], New York, 2002. MR 1903357
S. Pernet, X. Ferrieres, and G. Cohen, An original finite element method to solve Maxwell’s equations in time domain, Proceedings of EMC Zurich’2003, 18-20 February 2003, Zurich, Switzerland.
S. Prudhomme, F. Pascal, T. Oden, and A. Romkes, Review of a priori error estimation for Discontinuous Galerkin, Orsay, 2000, 2000-02.
Béatrice Rivière, Mary F. Wheeler, and Vivette Girault, A priori error estimates for finite element methods based on discontinuous approximation spaces for elliptic problems, SIAM J. Numer. Anal. 39 (2001), no. 3, 902–931. MR 1860450, DOI 10.1137/S003614290037174X
Christine Bernardi and Yvon Maday, Spectral methods, Handbook of numerical analysis, Vol. V, Handb. Numer. Anal., V, North-Holland, Amsterdam, 1997, pp. 209–485. MR 1470226, DOI 10.1016/S1570-8659(97)80003-8
D. Gottlieb and J.S. Hesthaven, Spectral methods for time-dependent problems, Cambridge Press.
A. Elmkies, Sur les éléments finis d’arête pour la résolution des équations de Maxwell en milieu anisotrope et pour des maillages quelconques, Université Paris IX-Dauphine, 1998, Thèse de mathématiques appliquées à l’ingéniérie.
Vivette Girault and Pierre-Arnaud Raviart, Finite element methods for Navier-Stokes equations, Springer Series in Computational Mathematics, vol. 5, Springer-Verlag, Berlin, 1986. Theory and algorithms. MR 851383, DOI 10.1007/978-3-642-61623-5
S. Pernet, Etude de méthodes d’ordre élevé pour résoudre les équations de Maxwell dans le domaine temporel. Application à la détection et à la compatibilité électromagnétique, Thesis, University of Paris, IX, November 2004.