A Discontinuous Galerkin method is used for to the numerical solution of the time-domain Maxwell equations on unstructured meshes. The method relies on the choice of local basis functions, a centered mean approximation for the surface integrals and a second-order leap-frog scheme for advancing in time. The method is proved to be stable for cases with either metallic or absorbing boundary conditions, for a large class of basis functions. A discrete analog of the electromagnetic energy is conserved for metallic cavities. Convergence is proved for Discontinuous elements on tetrahedral meshes, as well as a discrete divergence preservation property. Promising numerical examples with low-order elements show the potential of the method.
@article{M2AN_2005__39_6_1149_0, author = {Fezoui, Loula and Lanteri, St\'ephane and Lohrengel, St\'ephanie and Piperno, Serge}, title = {Convergence and stability of a discontinuous Galerkin time-domain method for the 3D heterogeneous Maxwell equations on unstructured meshes}, journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique}, volume = {39}, year = {2005}, pages = {1149-1176}, doi = {10.1051/m2an:2005049}, mrnumber = {2195908}, zbl = {1094.78008}, language = {en}, url = {http://dml.mathdoc.fr/item/M2AN_2005__39_6_1149_0} }
Fezoui, Loula; Lanteri, Stéphane; Lohrengel, Stéphanie; Piperno, Serge. Convergence and stability of a discontinuous Galerkin time-domain method for the 3D heterogeneous Maxwell equations on unstructured meshes. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 39 (2005) pp. 1149-1176. doi : 10.1051/m2an:2005049. http://gdmltest.u-ga.fr/item/M2AN_2005__39_6_1149_0/
[1] Resolution of the non-stationary or harmonic Maxwell equations by a discontinuous finite element method. Application to an E.M.I. (electromagnetic impulse) case. Comput. Method Appl. Sci. Engrg. (1991) 405-422.
, and ,[2] A discontinuous galerkin method for 3d maxwell's equation on non-conforming grids, in Sixth International Conference on Mathematical and Numerical Aspects of Wave Propagation. G.C. Cohen Ed., Springer, Jyvskyl, Finland (2003) 389-394. | Zbl 1075.78002
, and ,[3] The Finite Element Method for Elliptic Problems. North Holland-Elsevier Science Publishers, Amsterdam, New York, Oxford (1978). | MR 520174 | Zbl 0383.65058
,[4] A parallel FVTD Maxwell solver using 3D unstructured meshes, in 13th annual review of progress in applied computational electromagnetics, Monterey, California (1997) 359-365.
, , and ,[5] | MR 1842160 | Zbl 0935.00043
, and , Eds., Discontinuous Galerkin methods. Theory, computation and applications. Lect. Notes Comput. Sci. Eng. 11 (2000).[6] Locally divergence-free discontinuous galerkin methods for the maxwell equations. J. Comput. Phys. 194 (2004) 588-610. | Zbl 1049.78019
, and ,[7] Éléments finis d'arête et condensation de masse pour les équations de Maxwell: le cas de dimension 3. C. R. Acad. Sci. Paris Sér. I Math. 325 (1997) 1217-1222. | Zbl 0893.65068
and ,[8] The finite volume method, Handbook Numer. Anal., North Holland-Elsevier Science Publishers, Amsterdam, New York, Oxford 7-3 (2000). | MR 1804748 | Zbl 0981.65095
, and ,[9] Explicit finite element methods for symmetric hyperbolic equations. SIAM J. Numer. Anal. 36 (1998) 935-952. | Zbl 0923.65065
and ,[10] Stable spectral methods on tetrahedral elements. SIAM J. Sci. Comput. 21 (2000) 2352-2380. | Zbl 0959.65112
and ,[11] Nodal high-order methods on unstructured grids. I: Time-domain solution of Maxwell's equations. J. Comput. Phys. 181 (2002) 186-221. | Zbl 1014.78016
and ,[12] High-order nodal discontinuous galerkin methods for the maxwell eigenvalue problem. Philos. Trans. Roy. Soc. London Ser. A 362 (2004) 493-524. | Zbl 1078.78014
and ,[13] Mimetic discretizations for Maxwell's equations. J. Comput. Phys. 151 (1999) 881-909. | Zbl 0956.78015
and ,[14] A new second order 3D edge element on tetrahedra for time dependent Maxwell's equations, in Fifth International Conference on Mathematical and Numerical Aspects of Wave Propagation, A. Bermudez, D. Gomez, C. Hazard, P. Joly, J.-E. Roberts, Eds., SIAM, Santiago de Compostella, Spain (2000) 842-847. | Zbl 0995.78040
and ,[15] Discontinuous spectral element approximation of Maxwell's equations, in Discontinuous Galerkin methods. Theory, computation and applications., B. Cockburn and G.E. Karniadakis, C.-W. Shu, Eds. Lect. Notes Comput. Sci. Eng. 11 (2000) 355-362. | Zbl 0957.78023
, and ,[16] A Lax-Wendroff type theorem for upwind finite volume schemes in 2-D. J. Numer. Math. 4 (1996) 279-292. | Zbl 0872.65093
, and ,[17] A FV scheme for Maxwell's equations: Convergence analysis on unstructured meshes, in Finite Volumes for Complex Applications III, R. Herbin, D. Kröner, Eds., Hermes Penton Science, London, Porquerolles, France (2002) 219-226. | Zbl 1062.78017
and ,[18] Existence and uniqueness of the Maxwell's system solutions in heterogeneous and irregular media. C. R. Acad. Sci. Paris Sér. I Math. 330 (2000) 99-103. | Zbl 0942.78006
and ,[19] -stability of the upwind first order finite volume scheme for the Maxwell equation in two and three dimensions on arbitrary unstructured meshes. ESAIM: M2AN 34 (2000) 139-158. | Numdam | Zbl 0949.65104
,[20] Schémas en éléments finis discontinus localement raffinés en espace et en temps pour les équations de Maxwell 1D. INRIA Research report 4986 (2003).
,[21] A non-diffusive finite volume scheme for the 3D Maxwell equations on unstructured meshes. SIAM J. Numer. Anal. 39 (2002) 2089-2108. | Zbl pre01764395
, and ,[22] A new finite volume scheme for solving Maxwell's system. COMPEL 19 (2000) 913-931. | Zbl 0994.78021
,[23] A comparative study of characteristic-based algorithms for the Maxwell equations. J. Comput. Phys. 125 (1996) 378-394. | Zbl 0848.65087
and ,[24] Re-inventing electromagnetics: supercomputing solution of Maxwell's equations via direct time integration on space grids. AIAA paper 92-0333 (1992).
,[25] Convergence and error estimates in finite volume schemes for general multidimensional scalar conservation laws. I. Explicit monotone schemes. RAIRO Modél. Math. Anal. Numér. 28 (1994) 267-295. | Numdam | Zbl 0823.65087
,[26] Application of the discontinuous Galerkin method to Maxwell’s equations using unstructured polymorphic -finite elements, in Discontinuous Galerkin methods. Theory, computation and applications, B. Cockburn, G.E. Karniadakis, C.-W. Shu, Eds. Lect. Notes Comput. Sci. Eng. 11 (2000) 451-458. | Zbl 0957.78011
,[27] Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic media. IEEE T. Antenn. Prop. AP-16 (1966) 302-307.
,