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High Order Edge Elements for Electromagnetic Waves: Remarks on Numerical Dispersion

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Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2016

Part of the book series: Lecture Notes in Computational Science and Engineering ((LNCSE,volume 119))

Abstract

We recall one set of possible basis vector fields and two different sets of possible degrees of freedom, those related to “small-edges” and those defined by “moments”, for the Nédélec’s first family of high order edge elements. We thus address a dispersion analysis of the resulting methods, when the time-harmonic Maxwell’s equation for the electric field is discretized on a simplicial mesh.

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Notes

  1. 1.

    In two spatial dimensions, say x, y, we denote by ∂ x (resp. ∂ y ) the first order operator that associates any differentiable scalar function g with its partial derivative ∂ x g (resp. ∂ y g) w.r.t. the variable x (resp. y). For any vector field \(\mathbf{u} \in \mathbb{R}^{2}\), with u = (u x , u y )⊤, we define f = curl u such as the scalar function f = ∂ x u y − ∂ y u x ; for any scalar function v ∈ I​R, we define w = curl v such as the vector field w = (∂ y v, −∂ x v)⊤. Note that curl v = (grad v)⊥, where g = grad v = (∂ x v, ∂ y v)⊤.

  2. 2.

    The definition of Whitney forms relies on duality features from algebraic topology. Let w e be the Whitney edge forms, where e belongs to the set of mesh edges \(\mathcal{E}\). A field v, represented by the 1-form ∑ e v e w e, with v e = ∫ e v, and a curve γ, represented by the 1-chain ∑ e α e  e, with α e = ∫ γ w e, are in duality via the formula \(\int _{\gamma }\mathrm{v} =\sum _{e\in \mathcal{E}}\alpha _{e}\,\mathrm{v}_{e}\). What is meant by in duality is that ∫ γ v = 0 \(\ \forall \gamma\) implies v = 0 and the other way around. Duality stems from the property of edge elements, ∫ e′ w e = δ e′ e . Note that the same forms w e are involved in the description of both fields and curves: this duality is the key to understand why Whitney elements have the expression they have [3].

References

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Acknowledgements

The authors acknowledge the French National Research Agency (ANR) for its financial support (project MEDIMAX, ANR-13-MONU-0012). The last author warmly thanks the Università degli Studi di Verona for the possibility of studying as ERASMUS fellow at the Université Côte Azur in Nice.

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Authors and Affiliations

  1. Laboratoire J.A. Dieudonné, CNRS & Université Côte d’Azur, Parc Valrose, 06108, Nice Cedex 02, France

    Marcella Bonazzoli & Francesca Rapetti

  2. Laboratoire J.L. Lions, CNRS & Université Pierre et Marie Curie, B.P. 187, 4 Place Jussieu, 75252, Paris Cedex 05, France

    Pierre-Henri Tournier

  3. Università degli Studi di Verona, Via dell’Artigliere 8, 37129, Verona, Italy

    Chiara Venturini

Authors
  1. Marcella Bonazzoli
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  2. Francesca Rapetti
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  3. Pierre-Henri Tournier
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  4. Chiara Venturini
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Corresponding author

Correspondence to Marcella Bonazzoli .

Editor information

Editors and Affiliations

  1. School of Mechanical Engineering, University of Campinas, Campinas, Brazil

    Marco L. Bittencourt

  2. Department of Civil Engineering, Pontifical Catholic University of Rio de Janeiro, Rio de Janeiro, Brazil

    Ney A. Dumont

  3. EPFL-SB-MATHICSE-MCSS, Ecole Polytechnique Fédérale de Lausanne, Lausanne, Switzerland

    Jan S. Hesthaven

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Bonazzoli, M., Rapetti, F., Tournier, PH., Venturini, C. (2017). High Order Edge Elements for Electromagnetic Waves: Remarks on Numerical Dispersion. In: Bittencourt, M., Dumont, N., Hesthaven, J. (eds) Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2016. Lecture Notes in Computational Science and Engineering, vol 119. Springer, Cham. https://doi.org/10.1007/978-3-319-65870-4_14

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