Finite Element Heterogeneous Multiscale Method for Time-Dependent Maxwell’s Equations

  • Marlis Hochbruck
  • Christian StohrerEmail author
Conference paper
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 119)


We propose a Finite Element Heterogeneous Multiscale Method (FE-HMM) for time dependent Maxwell’s equations in second-order formulation in locally periodic materials. This method can approximate the effective behavior of an electromagnetic wave traveling through a highly oscillatory material without the need to resolve the microscopic details of the material. To prove an a-priori error bound for the semi-discrete FE-HMM scheme, we need a new generalization of a Strang-type lemma for second-order hyperbolic equations. Finally, we present a numerical example that is in accordance with the theoretical results.


Time dependent Maxwell’s equations Finite element heterogeneous multiscale method Strang-type lemma for hyperbolic PDEs 



We gratefully acknowledge financial support by the Deutsche Forschungsgemeinschaft (DFG) through CRC 1173 and the Klaus Tschira Stiftung. In addition we thank the anonymous referee for helpful suggestions.


  1. 1.
    A. Abdulle, The finite element heterogeneous multiscale method: a computational strategy for multiscale PDEs. GAKUTO Int. Ser. Math. Sci. Appl. 31, 133–181 (2009)zbMATHMathSciNetGoogle Scholar
  2. 2.
    A. Abdulle, M.J. Grote, Finite element heterogeneous multiscale method for the wave equation. Multiscale Model. Simul. 9(2), 766–792 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    A. Abdulle, W. E, B. Engquist, E. Vanden-Eijnden, The heterogeneous multiscale method. Acta Numer. 21, 1–87 (2012)Google Scholar
  4. 4.
    G. Allaire, Homogenization and two-scale convergence. SIAM J. Math. Anal. 23(6), 1482–1518 (1992)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    A. Bensoussan, J.L. Lions, G. Papanicolaou, Asymptotic analysis for periodic structures, in Studies in Mathematics and its Applications, vol. 5 (North-Holland Publishing Co., Amsterdam/New York, 1978)Google Scholar
  6. 6.
    V.T. Chu, V.H. Hoang, High-dimensional finite elements for multiscale Maxwell-type equations. IMA J. Numer. Anal. drx001 (2017, Online). doi:
  7. 7.
    P.G. Ciarlet, The finite element method for elliptic problems, in Classics in Applied Mathematics, vol. 40 (SIAM, Philadelphia, 2002)Google Scholar
  8. 8.
    P. Ciarlet Jr., J. Zou, Fully discrete finite element approaches for time-dependent Maxwell’s equations. Numer. Math. 82(2), 193–219 (1999)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    P. Ciarlet Jr., S. Fliss, C. Stohrer, On the approximation of electromagnetic fields by edge finite elements. Part 2: a heterogeneous multiscale method for Maxwell’s equations. Comput. Math. Appl. 73(9), 1900–1919 (2017)Google Scholar
  10. 10.
    L.C. Evans, Partial differential equations, Graduate Studies in Mathematics, vol. 19 (American Mathematical Society, Providence, 1998)Google Scholar
  11. 11.
    F. Hecht, New development in FreeFem++. J. Numer. Math. 20(3–4), 25–265 (2012)zbMATHMathSciNetGoogle Scholar
  12. 12.
    P. Henning, M. Ohlberger, B. Verfürth, A new heterogeneous multiscale method for time-harmonic Maxwell’s equations. SIAM J. Numer. Anal. 54(6), 3493–3522 (2016)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    V.V. Jikov, S.M. Kozlov, O.A. Oleinik, Homogenization of Differential Operators and Integral Functionals (Springer, Berlin, 1994)CrossRefGoogle Scholar
  14. 14.
    J.L. Lions, E. Magenes, Non-homogeneous boundary value problems and applications. Vol. I, Die Grundlehren der mathematischen Wissenschaften, vol. 181 (Springer, New York/Heidelberg, 1972)Google Scholar
  15. 15.
    P.A. Markowich, F. Poupaud, The Maxwell equation in a periodic medium: Homogenization of the energy density. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 23(2), 301–324 (1996)Google Scholar
  16. 16.
    P. Monk, Analysis of a finite element method for Maxwell’s equations. SIAM J. Numer. Anal. 29(3), 714–729 (1992)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    P. Monk, Finite Element Methods for Maxwell’s Equations (Oxford University Press, Oxford, 2003)CrossRefzbMATHGoogle Scholar
  18. 18.
    N. Wellander, Homogenization of the Maxwell equations: Case I. Linear theory. Appl. Math. 46(1), 29–51 (2001)zbMATHMathSciNetGoogle Scholar

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© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Institute of Applied and Numerical AnalysisKarlsruhe Institute of TechnologyKarlsruheGermany

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