Hostname: page-component-586b7cd67f-tf8b9 Total loading time: 0 Render date: 2024-11-26T11:05:54.688Z Has data issue: false hasContentIssue false
  • English
  • Français

Convergence and stability of a discontinuous Galerkin time-domain method for the 3D heterogeneous Maxwell equations on unstructured meshes

Published online by Cambridge University Press:  15 November 2005

Loula Fezoui ,
Stéphane Lanteri ,
Stéphanie Lohrengel  and
Serge Piperno
Loula Fezoui
Affiliation:
CERMICS, INRIA, BP93, 06902 Sophia-Antipolis Cedex, France. [email protected]
Stéphane Lanteri
Affiliation:
CERMICS, INRIA, BP93, 06902 Sophia-Antipolis Cedex, France. [email protected]
Stéphanie Lohrengel
Affiliation:
Dieudonné Lab., UNSA, UMR CNRS 6621, Parc Valrose, 06108 Nice Cedex 2, France.
Serge Piperno
Affiliation:
CERMICS, INRIA, BP93, 06902 Sophia-Antipolis Cedex, France. [email protected]
Get access

Abstract

A Discontinuous Galerkin method is used for to thenumerical solution of the time-domain Maxwell equations onunstructured meshes. The method relies on the choice of local basisfunctions, a centered mean approximation for the surface integralsand a second-order leap-frog scheme for advancing in time. The methodis proved to be stable for cases with either metallic or absorbingboundary conditions, for a large class of basis functions. A discrete analog of the electromagnetic energy is conserved formetallic cavities. Convergence is proved for $\mathbb{P}_k$ Discontinuous elements on tetrahedral meshes, as well as a discretedivergence preservation property. Promising numerical examples withlow-order elements show the potential of the method.

Keywords

Electromagneticsfinite volume methodsdiscontinuousGalerkin methodscentered fluxesleap-frog time schemeL 2 stabilityunstructured meshesabsorbing boundaryconditionconvergencedivergence preservation.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 39 , Issue 6 , November 2005 , pp. 1149 - 1176
Copyright
© EDP Sciences, SMAI, 2005

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

F. Bourdel, P.-A. Mazet and P. Helluy, 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.
N. Canouet, L. Fezoui and S. Piperno, 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.
P. Ciarlet, The Finite Element Method for Elliptic Problems. North Holland-Elsevier Science Publishers, Amsterdam, New York, Oxford (1978).
J.-P. Cioni, L. Fezoui, L. Anne and F. Poupaud, A parallel FVTD Maxwell solver using 3D unstructured meshes, in 13th annual review of progress in applied computational electromagnetics, Monterey, California (1997) 359–365.
B. Cockburn, G.E. Karniadakis and C.-W. Shu, Eds., Discontinuous Galerkin methods. Theory, computation and applications. Lect. Notes Comput. Sci. Eng. 11 (2000).
Cockburn, B., Li, F. and Shu, C.-W., Locally divergence-free discontinuous galerkin methods for the maxwell equations. J. Comput. Phys. 194 (2004) 588–610. CrossRef
Elmkies, A. and Joly, P., 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. CrossRef
R. Eymard, T. Gallouët and R. Herbin, The finite volume method, Handbook Numer. Anal., North Holland-Elsevier Science Publishers, Amsterdam, New York, Oxford 7-3 (2000).
Falk, R.S. and Richter, G.R., Explicit finite element methods for symmetric hyperbolic equations. SIAM J. Numer. Anal. 36 (1998) 935–952. CrossRef
Hesthaven, J. and Teng, C., Stable spectral methods on tetrahedral elements. SIAM J. Sci. Comput. 21 (2000) 2352–2380. CrossRef
Hesthaven, J. and Warburton, T., Nodal high-order methods on unstructured grids. I: Time-domain solution of Maxwell's equations. J. Comput. Phys. 181 (2002) 186–221. CrossRef
Hesthaven, J. and Warburton, T., High-order nodal discontinuous galerkin methods for the maxwell eigenvalue problem. Philos. Trans. Roy. Soc. London Ser. A 362 (2004) 493–524. CrossRef
Hyman, J.M. and Shashkov, M., Mimetic discretizations for Maxwell's equations. J. Comput. Phys. 151 (1999) 881–909. CrossRef
P. Joly and C. Poirier, 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.
Kopriva, D.A., Woodruff, S.L. and Hussaini, M.Y., 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. CrossRef
Kröner, D., Rokyta, M. and Wierse, M., Lax-Wendroff, A type theorem for upwind finite volume schemes in 2-D. J. Numer. Math. 4 (1996) 279–292.
S. Lohrengel and M. Remaki, 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.
Poupaud, F. and Remaki, M., 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. CrossRef
Piperno, S., L 2-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. CrossRef
S. Piperno, Schémas en éléments finis discontinus localement raffinés en espace et en temps pour les équations de Maxwell 1D. INRIA Research report 4986 (2003).
Piperno, S., Remaki, M. and Fezoui, L., A non-diffusive finite volume scheme for the 3D Maxwell equations on unstructured meshes. SIAM J. Numer. Anal. 39 (2002) 2089–2108. CrossRef
Remaki, M., A new finite volume scheme for solving Maxwell's system. COMPEL 19 (2000) 913–931. CrossRef
Shang, J. and Fithen, R., A comparative study of characteristic-based algorithms for the Maxwell equations. J. Comput. Phys. 125 (1996) 378–394. CrossRef
A. Taflove, Re-inventing electromagnetics: supercomputing solution of Maxwell's equations via direct time integration on space grids. AIAA paper 92–0333 (1992).
Convergence, J.-P. Vila 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.
Warburton, T., Application of the discontinuous Galerkin method to Maxwell's equations using unstructured polymorphic hp-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. CrossRef
K.S. Yee, Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic media. IEEE T. Antenn. Prop. AP-16 (1966) 302–307.