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The Maxwell’s System and Spurious Modes

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Finite Element and Discontinuous Galerkin Methods for Transient Wave Equations

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Abstract

This chapter begins with the construction of different approximations of Maxwell’s equations by finite element methods and the comparison of their performance versus discontinuous Galekin methods. Then, we treat the important and not obvious problem of spurious modes which appear in most approximations and we indicate how to suppress these modes. The last section contains error estimates of discontinuous Galekin methods with mass-lumping on hexahedra.

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Notes

  1. 1.

    One can multiply the second equation by \(\mu ^{-1}\) in order to obtain transposed stiffness matrices in the two equations.

  2. 2.

    Which is possible since \(\left[ H^1( \varOmega )\right] ^3 \subset H(\mathbf{curl},\varOmega )\). The boundary condition will be treated below.

  3. 3.

    For non-structured meshes, one can obtain less or more degrees of freedom for a vertex and a point on an edge.

  4. 4.

    Since they were made by different students, the values of the eigenmodes vary in the following figures, but they represent the same eigenmodes with a multiplying coefficient.

  5. 5.

    This judicious remark was made by I. Perugia during a workshop.

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Authors and Affiliations

  1. Inria, Saclay, France

    Gary Cohen

  2. The French Aerospace Lab, ONERA, Toulouse, France

    Sébastien Pernet

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Correspondence to Gary Cohen .

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Cohen, G., Pernet, S. (2017). The Maxwell’s System and Spurious Modes. In: Finite Element and Discontinuous Galerkin Methods for Transient Wave Equations. Scientific Computation. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-7761-2_5

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  • DOI: https://doi.org/10.1007/978-94-017-7761-2_5

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