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

  • Marcella BonazzoliEmail author
  • Francesca Rapetti
  • Pierre-Henri Tournier
  • Chiara Venturini
Conference paper
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 119)


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.



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.


  1. 1.
    M. Bonazzoli, F. Rapetti, High order finite elements in numerical electromagnetism: degrees of freedom and generators in duality. Numer. Algorithms 74(1), 111–136 (2017)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    M. Bonazzoli, V. Dolean, F. Rapetti, P.-H. Tournier, Parallel preconditioners for high order discretizations arising from full system modeling for brain microwave imaging. Int. J. Numer. Modell. Electron. Netw. Devices Fields. doi:10.1002/jnm.2229 [math.NA] (2017, in press)
  3. 3.
    A. Bossavit, Computational Electromagnetism (Academic, New York, 1998)zbMATHGoogle Scholar
  4. 4.
    A. Bossavit, J.C. Vérité, A mixed FEM-BIEM method to solve eddy-current problems. IEEE Trans. Magn. 18, 431–435 (1982)CrossRefGoogle Scholar
  5. 5.
    S.H. Christiansen, F. Rapetti, On high order finite element spaces of differential forms. Math. Comput. 85(298), 517–548 (2016)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    G.E. Karniadakis, S.J. Sherwin, Spectral/hp Element Methods for CFD (Oxford University Press, New York, 1999)zbMATHGoogle Scholar
  7. 7.
    I. Mazzieri, F. Rapetti, Dispersion analysis of triangle-based spectral element methods for elastic wave propagation. Numer. Algorithms 60, 631–650 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    P. Monk, Finite Element Methods for Maxwell’s Equations. (Oxford Science Publications, New York, 2003)Google Scholar
  9. 9.
    J.-C. Nédélec, Mixed finite elements in R 3. Numer. Math. 35, 315–341 (1980)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    F. Rapetti, High order edge elements on simplicial meshes. Meth. Math. en Anal. Num. 41(6), 1001–1020 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    F. Rapetti, A. Bossavit, Whitney forms of higher degree. SIAM J. Numer. Anal. 47(3), 2369–2386 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    G.S. Warren, W.R. Scott, Numerical dispersion in the finite-element method using triangular edge elements. Microw. Opt. Technol. Lett. 9(6), 315–319 (Wiley, New York, 1995)Google Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Marcella Bonazzoli
    • 1
    Email author
  • Francesca Rapetti
    • 1
  • Pierre-Henri Tournier
    • 2
  • Chiara Venturini
    • 3
  1. 1.Laboratoire J.A. DieudonnéCNRS & Université Côte d’AzurNice Cedex 02France
  2. 2.Laboratoire J.L. LionsCNRS & Université Pierre et Marie CurieParis Cedex 05France
  3. 3.Università degli Studi di VeronaVeronaItaly

Personalised recommendations