Advances in Computational Mathematics

, Volume 22, Issue 1, pp 21–48 | Cite as

Finite element analysis of transient electromagnetic scattering problems

  • Tri Van
  • Aihua Wood


In this paper, Newmark time-stepping scheme and edge elements are used to numerically solve the time-dependent scattering problem in a three-dimensional cavity. Finite element methods based on the variational formulation derived in [23] are considered. Due to the lack of regularity ofε r , the existence and uniqueness of the discrete solutions and their convergence are proved by using the concept of collectively compact operators. An optimal convergence rate in the energy norm is also established.


Maxwell’s equations finite element methods Newmark scheme error estimates stability cavity 

AMS subject classification

35L05 65M60 78M10 74J20 


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  1. [1]
    T. Abboud and T. Sayah, Potentiels retardés pour les obstacles homogènes par morceaux, Rapport interne au CMAP, École Polytechnique (1998).Google Scholar
  2. [2]
    A. Alonso and A. Valli, An optimal domain decomposition preconditioner for low-frequency timeharmonic Maxwell’s equations, Math. Comp. 68 (1999) 607–631.zbMATHMathSciNetGoogle Scholar
  3. [3]
    H. Ammari, G. Bao and A. Wood, Analysis of the electromagnetic scattering from a cavity, Japan J. Industr. Appl. Math. 19 (2001) 301–308.CrossRefMathSciNetGoogle Scholar
  4. [4]
    H. Ammari, G. Bao and A. Wood, A cavity problem for Maxwell’s equations, Methods Math. Appl., to appear.Google Scholar
  5. [5]
    C. Amrouche, C. Bernardi, M. Dauge and V. Girault, Vector potentials in three-dimensional non-smooth domains, Math. Methods Appl. Sci. 21 (1998).Google Scholar
  6. [6]
    S. Caorsi, P. Fernandes and M. Raffetto, On the convergence of Galerkin finite element approximations of electromagnetic eigenproblems, SIAM J. Numer. Anal. 38 (2000) 580–607.zbMATHMathSciNetGoogle Scholar
  7. [7]
    Z. Chen, Q. Du and J. Zou, Finite element methods with matching and nonmatching meshes for Maxwell’s equations with discontinuous coefficients, SIAM J. Numer. Anal. 37 (2000) 1542–1570.zbMATHMathSciNetGoogle Scholar
  8. [8]
    P. Ciarlet and J. Zou, Fully discrete finite element approaches for time-dependent Maxwell’s equations, Numer. Math. 82 (1999) 193–219.zbMATHMathSciNetGoogle Scholar
  9. [9]
    M. Costabel, A coercive bilinear form for Maxwell’s equations, J. Math. Anal. Appl. 157 (1991) 527–541.zbMATHMathSciNetGoogle Scholar
  10. [10]
    M. Costabel and M. Dauge, Singularities of electromagnetic fields in polyhedral domains, Technical Report, IRMAR, Universitá de Rennes 1, France (1997); available at∼costabel/.Google Scholar
  11. [11]
    M. Eller, V. Isakov, G. Nakamura and D. Tataru,Uniqueness and Stability in the Cauchy Problem for Maxwell’s and Elasticity Systems, eds. D. Cioranescu and J. L. Lions, College de France Seminar, Research Notes in Mathematics, Vol. 14 (Chapman & Hill/CRC, New York) to appear.Google Scholar
  12. [12]
    V. Girault and P.A. Raviart,Finite Element Methods for Navier-Stokes Equations. Theory and Algorithms (Springer, New York, 1986).zbMATHGoogle Scholar
  13. [13]
    D. Givoli, Non reflecting boundary conditions, J. Comput. Phys. 94 (1991) 1–29.zbMATHMathSciNetGoogle Scholar
  14. [14]
    C. Hazard and M. Lenoir, On the solution of time-harmonic scattering problems for Maxwell’s equations, SIAM J. Math. Anal. 27 (1996) 1597–1630.zbMATHMathSciNetGoogle Scholar
  15. [15]
    F. Kikuchi, On a discrete compactness property for the Nédélec finite elements, J. Fac. Sci. Univ. Tokyo, Sect. IA Math. 36 (1989) 479–490.zbMATHMathSciNetGoogle Scholar
  16. [16]
    A. Kirsch and P. Monk, A finite element method for approximating electro-magnetic scattering from a conducting object, Preprint (2000).Google Scholar
  17. [17]
    K.S. Komisarek, N.N. Wang, A.K. Dominek and R. Hann, An investigation of new FETD/ABC methods of computation of scattering from three-dimensional material objects, IEEE Trans. Antennas Propagation 47 (1999) 1579–1585.Google Scholar
  18. [18]
    R. Kress,Linear Integral Equations, 2nd ed. (Springer, New York, 1999).zbMATHGoogle Scholar
  19. [19]
    P. Monk, Analysis of a finite element method for Maxwell’s equations, SIAM J. Numer. Anal. 29 (1992) 714–729.zbMATHMathSciNetGoogle Scholar
  20. [20]
    P.B. Monk, A mixed method for approximating Maxwell’s equations, SIAM J. Numer. Anal. 28 (1991) 1610–1634.zbMATHMathSciNetGoogle Scholar
  21. [21]
    J.C. Nédélec, Mixed finite elements in ℝ3, Numer. Math. 35 (1980) 315–341.zbMATHMathSciNetGoogle Scholar
  22. [22]
    M. Taylor,Pseudodifferential Operators and Nonlinear PDE (Birkhäuser, Boston, 1991).zbMATHGoogle Scholar
  23. [23]
    T. Van and A. Wood, Analysis of time-domain Maxwell’s equations for 3-D electromagnetic cavities, Adv. Comput. Math. 16 (2002) 211–228.zbMATHMathSciNetGoogle Scholar
  24. [24]
    T. Van and A. Wood, A time-domain finite element method for 2-D cavities, J. Comput. Phys. 183 (2002) 486–507.zbMATHMathSciNetGoogle Scholar
  25. [25]
    T. Van and A. Wood, A time-marching finite element method for an electromagnetic scattering problem, Methods Math. Appl., to appear.Google Scholar
  26. [26]
    V. Vogelsang, On the strong unique continuation principle for inequalities of Maxwell type, Math. Ann. 289 (1991) 285–295.zbMATHMathSciNetGoogle Scholar
  27. [27]
    C. Weber, Regularity theorems for Maxwell’s equations, Math. Methods Appl. Sci. 3 (1981) 523–536.zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2005

Authors and Affiliations

  • Tri Van
    • 1
  • Aihua Wood
    • 2
  1. 1.Mission Research CorporationDaytonUSA
  2. 2.Air Force Institute of TechnologyAFIT/ENC, WPAFBUSA

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