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Finite element analysis of transient electromagnetic scattering problems

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Abstract

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 ofεr, 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.

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References

  1. T. Abboud and T. Sayah, Potentiels retardés pour les obstacles homogènes par morceaux, Rapport interne au CMAP, École Polytechnique (1998).

  2. A. Alonso and A. Valli, An optimal domain decomposition preconditioner for low-frequency timeharmonic Maxwell’s equations, Math. Comp. 68 (1999) 607–631.

    Article  MathSciNet  Google Scholar 

  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.

    Article  MathSciNet  Google Scholar 

  4. H. Ammari, G. Bao and A. Wood, A cavity problem for Maxwell’s equations, Methods Math. Appl., to appear.

  5. C. Amrouche, C. Bernardi, M. Dauge and V. Girault, Vector potentials in three-dimensional non-smooth domains, Math. Methods Appl. Sci. 21 (1998).

  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.

    Article  MathSciNet  Google Scholar 

  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.

    Article  MathSciNet  Google Scholar 

  8. P. Ciarlet and J. Zou, Fully discrete finite element approaches for time-dependent Maxwell’s equations, Numer. Math. 82 (1999) 193–219.

    Article  MathSciNet  Google Scholar 

  9. M. Costabel, A coercive bilinear form for Maxwell’s equations, J. Math. Anal. Appl. 157 (1991) 527–541.

    Article  MathSciNet  Google Scholar 

  10. M. Costabel and M. Dauge, Singularities of electromagnetic fields in polyhedral domains, Technical Report, IRMAR, Universitá de Rennes 1, France (1997); available at http://www.maths.univrennes1.fr/∼costabel/.

    MATH  Google Scholar 

  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.

  12. V. Girault and P.A. Raviart,Finite Element Methods for Navier-Stokes Equations. Theory and Algorithms (Springer, New York, 1986).

    Book  Google Scholar 

  13. D. Givoli, Non reflecting boundary conditions, J. Comput. Phys. 94 (1991) 1–29.

    Article  MathSciNet  Google Scholar 

  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.

    Article  MathSciNet  Google Scholar 

  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.

    MathSciNet  MATH  Google Scholar 

  16. A. Kirsch and P. Monk, A finite element method for approximating electro-magnetic scattering from a conducting object, Preprint (2000).

  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.

    Article  Google Scholar 

  18. R. Kress,Linear Integral Equations, 2nd ed. (Springer, New York, 1999).

    Book  Google Scholar 

  19. P. Monk, Analysis of a finite element method for Maxwell’s equations, SIAM J. Numer. Anal. 29 (1992) 714–729.

    Article  MathSciNet  Google Scholar 

  20. P.B. Monk, A mixed method for approximating Maxwell’s equations, SIAM J. Numer. Anal. 28 (1991) 1610–1634.

    Article  MathSciNet  Google Scholar 

  21. J.C. Nédélec, Mixed finite elements in ℝ3, Numer. Math. 35 (1980) 315–341.

    Article  MathSciNet  Google Scholar 

  22. M. Taylor,Pseudodifferential Operators and Nonlinear PDE (Birkhäuser, Boston, 1991).

    Book  Google Scholar 

  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.

    Article  MathSciNet  Google Scholar 

  24. T. Van and A. Wood, A time-domain finite element method for 2-D cavities, J. Comput. Phys. 183 (2002) 486–507.

    Article  MathSciNet  Google Scholar 

  25. T. Van and A. Wood, A time-marching finite element method for an electromagnetic scattering problem, Methods Math. Appl., to appear.

  26. V. Vogelsang, On the strong unique continuation principle for inequalities of Maxwell type, Math. Ann. 289 (1991) 285–295.

    Article  MathSciNet  Google Scholar 

  27. C. Weber, Regularity theorems for Maxwell’s equations, Math. Methods Appl. Sci. 3 (1981) 523–536.

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Mission Research Corporation, 3975 Research Blvd, 45430, Dayton, OH, USA

    Tri Van

  2. Air Force Institute of Technology, AFIT/ENC, WPAFB, 2950 P St., 45433, OH, USA

    Aihua Wood

Authors
  1. Tri Van
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  2. Aihua Wood
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Communicated by C.A. Micchelli

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Van, T., Wood, A. Finite element analysis of transient electromagnetic scattering problems. Adv Comput Math 22, 21–48 (2005). https://doi.org/10.1007/s10444-005-1176-x

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  • DOI: https://doi.org/10.1007/s10444-005-1176-x

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