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A-Priori Convergence Analysis of a Discontinuous Galerkin Time-Domain Method to Solve Maxwell’s Equations on Hybrid Meshes

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Numerical Mathematics and Advanced Applications 2011

Abstract

We study a multi-element Discontinuous Galerkin Time Domain (DGTD) method for solving the system of unsteady Maxwell equations. This method is formulated on a non-conforming and hybrid mesh combining a structured (orthogonal, large size elements) quadrangulation of the regular zones of the computational domain with an unstructured triangulation for the discretization of the irregularly shaped objects. The main objective is to enhance the flexibility and the efficiency of DGTD methods. Within each element, the electromagnetic field components are approximated by a high order nodal polynomial, using a centered flux for the surface integrals and a second order Leap-Frog scheme for the time integration of the associated semi-discrete equations. We formulate the 3D discretization scheme, present the results of mathematical analysis (L 2 stability and a-priori convergence in 3D). Finally, the 2D numerical performance and convergence is demonstrated.

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Acknowledgements

The authors gratefully acknowledge support from Région Ile-de-France in the framework of the MIEL3D-MESHER project of the System@tic Paris-Région cluster.

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Authors and Affiliations

  1. Nachos project-team, INRIA Sophia Antipolis – Méditerranée research center 2004 Route des Lucioles, 93, 06902, Sophia Antipolis Cedex, France

    C. Durochat

  2. Jean-Alexandre Dieudonné Mathematics Laboratory, University of Nice – Sophia Antipolis, 06108, Nice Cedex 02, France

    C. Scheid

Authors
  1. C. Durochat
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  2. C. Scheid
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Corresponding author

Correspondence to C. Durochat .

Editor information

Editors and Affiliations

  1. , Department of Mathematics, University of Leicester, Leicester, LE!1 7RH, United Kingdom

    Andrea Cangiani

  2. , Department of Mathematics, University of Leicester, Leicester, LE1 7RL, United Kingdom

    Ruslan L. Davidchack

  3. , Department of Mathematics, University of Leicester, Leicester, LE1 7RH, United Kingdom

    Emmanuil Georgoulis

  4. , Department of Mathematics, University of Leicester, Leicester, LE1 7RH, United Kingdom

    Alexander N. Gorban

  5. , Department of Mathematics, University of Leicester, Leicester, LE1 7RH, United Kingdom

    Jeremy Levesley

  6. , School of Mathematical Sciences, University of Nottingham, University Park, Nottingham, NG7 2RD, United Kingdom

    Michael V. Tretyakov

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Durochat, C., Scheid, C. (2013). A-Priori Convergence Analysis of a Discontinuous Galerkin Time-Domain Method to Solve Maxwell’s Equations on Hybrid Meshes. In: Cangiani, A., Davidchack, R., Georgoulis, E., Gorban, A., Levesley, J., Tretyakov, M. (eds) Numerical Mathematics and Advanced Applications 2011. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-33134-3_10

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