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Convergence of Explicit \(P_1\) Finite-Element Solutions to Maxwell’s Equations

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Mathematical and Numerical Approaches for Multi-Wave Inverse Problems (CIRM 2019)

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Abstract

This paper is devoted to the numerical validation of an explicit finite-difference scheme for the integration in time of Maxwell’s equations in terms of the sole electric field. The space discretization is performed by the standard \(P_1\) finite element method assorted with the treatment of the time-derivative term by a technique of the mass-lumping type. The rigorous reliability analysis of this numerical model was the subject of authors’ another paper [2]. More specifically such a study applies to the particular case where the electric permittivity has a constant value outside a sub-domain, whose closure does not intersect the boundary of the domain where the problem is defined. Our numerical experiments in two-dimension space certify that the convergence results previously derived for this approach are optimal, as long as the underlying CFL condition is satisfied.

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Acknowledgements

The research of the first author is supported by the Swedish Research Council grant VR 2018-03661. The second author gratefully acknowledges the financial support provided by CNPq/Brazil through grant 307996/2008-5.

Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, Chalmers University of Technology and University of Gothenburg, SE-412 96, Gothenburg, Sweden

    Larisa Beilina

  2. Institut Jean Le Rond d’Alembert, UMR 7190 CNRS - Sorbonne Universit é, 75005, Paris, France

    V. Ruas

  3. CNPq, Brasilia, Brazil

    V. Ruas

Authors
  1. Larisa Beilina
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  2. V. Ruas
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Corresponding author

Correspondence to Larisa Beilina .

Editor information

Editors and Affiliations

  1. Department of Mathematical Sciences, Chalmers University of Technology and University of Gothenburg, Gothenburg, Sweden

    Larisa Beilina

  2. Départment de Mathématiques - MAPMO, University of Orléans, Orleans, France

    Maïtine Bergounioux

  3. Institut de Mathématiques de Marseille, Aix-Marseille University, Marseille, France

    Michel Cristofol

  4. Institut Fresnel, Marseille, France

    Anabela Da Silva

  5. Institut Fresnel, Marseille, France

    Amelie Litman

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Beilina, L., Ruas, V. (2020). Convergence of Explicit \(P_1\) Finite-Element Solutions to Maxwell’s Equations. In: Beilina, L., Bergounioux, M., Cristofol, M., Da Silva, A., Litman, A. (eds) Mathematical and Numerical Approaches for Multi-Wave Inverse Problems. CIRM 2019. Springer Proceedings in Mathematics & Statistics, vol 328. Springer, Cham. https://doi.org/10.1007/978-3-030-48634-1_7

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