Introduction to Applications of Numerical Analysis in Time Domain Computational Electromagnetism

  • Qiang Chen
  • Peter MonkEmail author
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 85)


We discuss two techniques for the solution of the time domain Maxwell system. The first is a partial differential equation based approach using conforming finite elements and implicit time stepping that is suitable when stiff problems are encountered, and where the medium is inhomogeneous. In particular we analyze the use of edge elements and certain A-stable schemes using the Fourier-Laplace transform. For a homogeneous medium, an integral equation approach can be used and we describe and analyze the convolution quadrature method applied to the electric field integral equation. In either case we emphasize that the convergence analysis depends on energy estimates for the continuous problem.


Boundary Integral Equation Discontinuous Galerkin Discontinuous Galerkin Method Finite Difference Time Domain Edge Element 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



Our research is supported in part by a grant from NSF (DMS-0811104). PM would like to thank BICOM at Brunel University, UK for a visiting position during the writing of most of this paper.


  1. 1.
    T. Abboud, J.-C. Nédélec, and J. Volakis, Stable solution of the retarded potential equations, in Proc. 17th Ann. Rev. Progress in Appl. Comp. Electromagnetics, Monterey, CA, 2001, pp. 146–151.Google Scholar
  2. 2.
    J. Adam, A. Serveniere, J. Nédélec, and P. Raviart, Study of an implicit scheme for integrating Maxwell’s equations, Comput. Meth. Appl. Mech. Eng., 22 (1980), pp. 327–46.Google Scholar
  3. 3.
    A. Aimi, M. Diligenti, C. Guardasoni, I. Mazzieri, and S. Panizzi, An energy approach to spacetime Galerkin BEM for wave propagation problems, Int. J. Numer. Meth. Eng., DOI:10.1002/nme.2660 (2009).Google Scholar
  4. 4.
    M. Ainsworth, Dispersive and dissipative behaviour of high order discontinuous Galerkin finite element methods, J. Comp. Phys., 198 (2004), pp. 106–130.Google Scholar
  5. 5.
    M. Ainsworth, Dispersive properties of high-order Nedelec/edge element approximation of the time-harmonic Maxwell equations, Phil. Trans. Roy. Soc. A, 362 (2004), pp. 471–91.Google Scholar
  6. 6.
    M. Ainsworth and J. Coyle, Hierarchic finite element bases on unstructured tetrahedral meshes, Int. J. Numer. Meth. Eng., 58 (2003), pp. 2103–30.Google Scholar
  7. 7.
    M. Ainsworth, P. Monk, and W. Muniz, Dispersive and dissipative properties of discontinuous Galerkin methods for the wave equation, J. Sci. Comput., 27 (2006), pp. 5–40.Google Scholar
  8. 8.
    M. Ainsworth and H. Wajid, Optimally blended spectral-finite element scheme for wave propagation and nonstandard reduced integration, SIAM J. Numer. Anal., 48 (2010), pp. 346–71.Google Scholar
  9. 9.
    C. Amrouche, C. Bernardi, M. Dauge, and V. Girault, Vector potentials in three-dimensional nonsmooth domains, Math. Meth. Appl. Sci., 21 (1998), pp. 823–64.Google Scholar
  10. 10.
    D. Arnold, R. Falk, and R. Winther, Finite element exterior calculus, homological techniques, and applications, Acta Numer., 15 (2006), pp. 1–155.Google Scholar
  11. 11.
    D. Arnold, R. Falk, and R. Winther, Finite element exterior calculus from Hodge theory to numerical stability, Bulletin of the American Mathematical Society, 47 (2010), pp. 281–354.Google Scholar
  12. 12.
    D. Arnold, R. Falk, and R. Winthur, Multigrid in H(div) and H(curl), Numer. Math., 85 (2000), pp. 197–217.Google Scholar
  13. 13.
    F. Assous, P. Degond, E. Heintze, P. Raviart, and J. Segreé, On a finite-element method for solving the three-dimensional Maxwell equations, J. Comput. Phys., 109 (1993), pp. 222–37open.Google Scholar
  14. 14.
    F. Assous, P. Degond, and J. Segré, Numerical approximation of the Maxwell equations in inhomogeneous media by a p 1 conforming finite element method, J. Comput. Phys., 128 (1996), pp. 363–80.Google Scholar
  15. 15.
    F. Assous and M. Mikhaeli, Nitsche type method for approximating boundary conditions in the static maxwell equations, in Proceedings of the 26th IASTED International Conference on Modelling, Identification, and Control, MIC ’07, Anaheim, CA, USA, 2007, ACTA Press, pp. 402–407.Google Scholar
  16. 16.
    I. Babuška and S. Sauter, Is the pollution effect of the FEM avoidable for the Helmholtz equation considering high wave numbers?, SIAM J. Numer. Anal., 34 (1997), pp. 2392–423.Google Scholar
  17. 17.
    A. Bachelot and A. Pujols, Equations intégrales espace-temps pour le systeème de Maxwell, C.R. Acad. Sc. Paris, Série I, 314 (1992), pp. 639–44.Google Scholar
  18. 18.
    C. Balanis, Advanced Engineering Electromagnetics, Wiley, 1989.Google Scholar
  19. 19.
    A. Bamberger and T. Ha-Duong, Formulation variationnelle espace-temps pour le calcul par potentiel retarde de la diffraction d’une onde acoustique(i), Math. Mech. in the Appl. Sci, 8 (1986), pp. 405–435.Google Scholar
  20. 20.
    L. Banjai, Multistep and multistage convolution quadrature for the wave equation: Algorithms and experiments, SIAM J. Sci. Comput., 32 (2010), pp. 2964–2994.Google Scholar
  21. 21.
    L. Banjai and W. Hackbusch, Hierarchical matrix techniques for low- and high-frequency Helmholtz problems, IMA J. Numer. Anal., 28 (2008), pp. 46–79.Google Scholar
  22. 22.
    L. Banjai and S. Sauter, Rapid solution of the wave equation in unbounded domains, SIAM J. Numer. Anal., 47 (2008), pp. 227–49.Google Scholar
  23. 23.
    R. Barthelmé, P. Ciarlet, and E. Sonnendrücker, Generalized formulations of Maxwell’s equations for numerical Vlasov-Maxwell simulations, Math. Meth. Appl. Sci., 17 (2007), pp. 659–80.Google Scholar
  24. 24.
    A. Bendali, Numerical analysis of the exterior boundary value problem for the time harmonic Maxwell equations by a boundary finite element method. Part ii: The discrete problem, Math. Comput., 43 (1984), pp. 47–68.Google Scholar
  25. 25.
    J. Bérenger, A perfectly matched layer for the absorption of electromagnetics waves, J. Comput. Phys., 114 (1994), pp. 185–200.Google Scholar
  26. 26.
    H. Bertram, Theory of Magnetic Recording, Cambridge University Press, ber94.Google Scholar
  27. 27.
    D. Biskamp, An Introduction to Magnetohydrodynamics, Cambridge University Press, 2001.Google Scholar
  28. 28.
    D. Boffi, L. Gastaldi, and A. Buffa, Convergence analysis for hyperbolic evolution problems in mixed form, Tech. Rep. 17-PV, I.M.A.T.I.-C.N.R., University of Pavia, Italy, 2005.Google Scholar
  29. 29.
    A. Bossavit, Computational Electromagnetism, Academic Press, San Diego, 1998.Google Scholar
  30. 30.
    F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods, Springer, New York, 1991.Google Scholar
  31. 31.
    A. Buffa and S. Christiansen, A dual finite element complex on the barycentric refinement, Math. Comput., 76 (2007), pp. 1743–69.Google Scholar
  32. 32.
    A. Buffa and P. Ciarlet Jr., On traces for functional spaces related to Maxwell’s equations. I. An integration by parts formula in Lipschitz polyhedra, Math. Meth. Appl. Sci., 24 (2001), pp. 9–30.Google Scholar
  33. 33.
    A. Buffa and P. Ciarlet Jr., On traces for functional spaces related to Maxwell’s equations. II. Hodge decompositions on the boundary of Lipschitz polyhedra, Math. Meth. Appl. Sci., 24 (2001), pp. 31–48.Google Scholar
  34. 34.
    A. Buffa, M. Costabel, and D. Sheen, On the traces of H(curl,Ω) in Lipschitz domains, J. Math. Anal. Appl., 276 (2003), pp. 845–67.Google Scholar
  35. 35.
    A. Buffa, R. Hiptmair, T. von Petersdorff, and C. Schwab, Boundary element methods for Maxwell transmission problems in Lipschitz domains, Numer. Math., 95 (2003), pp. 459–85.Google Scholar
  36. 36.
    A. Buffa, P. Houston, and I. Perugia, Discontinuous Galerkin computation of the Maxwell eigenvalues on simplicial meshes, J. Comput. Appl. Math., 204 (2007), pp. 317–33.Google Scholar
  37. 37.
    A. Buffa and I. Perugia, Discontinuous galerkin approximation of the Maxwell eigenproblem, SIAM J. Numer. Anal., 44 (2006), pp. 2198–226.Google Scholar
  38. 38.
    A. Buffa, G. Sangalli, and R. Vazquez, Isogeometric analysis in electromagnetics: B-splines approximation, Comput. Meth. Appl. Mech. Eng., 199 (2010), pp. 1143–52.Google Scholar
  39. 39.
    A. Campbell, An introduction to numerical methods in superconductors, J. Supercond. Nov. Magn., (2010). DOI 10.1007/s10948-010-0895-5.Google Scholar
  40. 40.
    Q. Chen, P. Monk, D. Weile, and X. Wang, Analysis of convolution quadrature applied to the time-domain electric field integral equation. to appear in Communications in Computational Physics.Google Scholar
  41. 41.
    W. C. Chew and W. H. Weedon, A 3D perfectly matched medium from modified Maxwell’s equations with stretched coordinates, Microwave Opt. Technol. Lett., 7 (1994), pp. 599–604.Google Scholar
  42. 42.
    S. Christiansen, Foundations of finite element methods for wave equations of Maxwell type, in Applied Wave Mathematics, E. Quak and T. Soomere, eds., Springer-Verlag, Berlin Heidelberg, 2009, pp. 335–93.Google Scholar
  43. 43.
    P. Ciarlet, The Finite Element Method for Elliptic Problems, vol. 4 of Studies In Mathematics and Its Applications, North-Holland, New York, 1978.Google Scholar
  44. 44.
    P. Ciarlet and E. Jamelot, Continuous Galerkin methods for solving the time-dependent Maxwell equations in 3D geometries, J. Comput. Phys., 226 (2007), pp. 1122–35.Google Scholar
  45. 45.
    P. Ciarlet and J. Zou, Fully discrete finite element approaches for time-dependent Maxwell’s equations, Numer. Math., 82 (1999), pp. 193–219.Google Scholar
  46. 46.
    B. Cockburn and C.-W. Shu, The Runge-Kutta discontinuous Galerkin method for conservation laws V: Multidimensional systems, J. Comput. Phys., 141 (1998), pp. 199–224.Google Scholar
  47. 47.
    G. Cohen, Higher-order numerical methods for transient wave equations, Springer, Berlin, 2002.Google Scholar
  48. 48.
    G. Cohen, P. Joly, J. Roberts, and N. Tordjman, Higher order triangular finite elements with mass lumping for the wave equation, SIAM J. Numer. Anal., 38 (2001), pp. 2047–78.Google Scholar
  49. 49.
    G. Cohen and P. Monk, Gauss point mass lumping schemes for Maxwell’s equations, Numer. Meth. Partial Diff. Eqns., 14 (1998), pp. 63–88.Google Scholar
  50. 50.
    D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, Springer-Verlag, New York, 2nd ed., 1998.Google Scholar
  51. 51.
    K. Cools, F. Andriulli, F. Olyslager, and E. Michielssen, Time domain Calderón identities and their application to the integral equation analysis of scattering by PEC objects Part I: Preconditioning, IEEE Trans. Antennas Propagat., 57 (2009), pp. 2352–64.Google Scholar
  52. 52.
    M. Costabel, Fundamentals, vol. 1 of Encyclopedia of Computational Mechanics, John Wiley & Sons, 2004, ch. Time-dependent Problems with the Boundary Integral Equation Method.Google Scholar
  53. 53.
    P. Davies, Numerical stability and convergence of approximations of retarded potential integral equations, SIAM J. Numer. Anal., 31 (1994), pp. 856–75.Google Scholar
  54. 54.
    P. Davies and D. Duncan, Averaging techniques for time-marching schemes for retarded potential integral equations, Applied Numerical Mathematics, 23 (1997), pp. 291–310.Google Scholar
  55. 55.
    L. Demkowicz, hp-adaptive finite elements for time-harmonic Maxwell equations, in Topics in Computational Wave Propagation: Direct and Inverse Problems, M. Ainsworth, P. Davies, D. Duncan, P. Martin, and B. Rynne, eds., vol. 31 of Lecture Notes in Computational Science and Engineering, Springer, 2003.Google Scholar
  56. 56.
    L. Demkowicz, P. Monk, and L. Vardapetyan, de Rham diagram for hp finite element spaces, Comput. Math. Appl., 39 (2000), pp. 29–38.Google Scholar
  57. 57.
    G. Duvaut and J.-L. Lions, Inequalities in Mechanics and Physics, Springer, New York, 1976.Google Scholar
  58. 58.
    A. Elmkies and P. Joly, Elements finis d’arete et condensation de masse pour les equations de Maxwell: le cas 3D, C. R. Acad. Sci. Paris, Série 1, 325 (1997), pp. 1217–22.Google Scholar
  59. 59.
    B. Engquist and A. Majda, Absorbing boundary conditions for the numerical simulation of waves, Math. Comput., 31 (1977), pp. 629–51.Google Scholar
  60. 60.
    A. Ergin, B. Shanker, and E. Michielssen, The plane-wave time-domain algorithm for fast analysis of transient phenomena, IEEE Antennas and Propagation Magazine, 41 (1999), pp. 39–52.Google Scholar
  61. 61.
    L. Fezoui, S. Lanteri, S. Lohrengel, and S. Piperno, Convergence and stability of a discontinuous Galerkin time-domain method for the 3D heterogeneous Maxwell equations on unstructured meshes, ESAIM: Mathematical Modeling and Numerical Analysis, 39 (2005), pp. 1149–76.Google Scholar
  62. 62.
    A. Fisher, R. Rieben, G. Rodrigue, and D. White, A generalized mass lumping technique for vector finite-element solutions of the time-dependent maxwell equations, IEEE Trans. Antennas Propagat., 53 (2005), pp. 2900–10.Google Scholar
  63. 63.
    M. Gander, F. Magoulès, and F. Nataf, Optimized Schwarz methods without overlap for the Helmholtz equation, SIAM J. Sci. Comput., 24 (2002), pp. 36–80.Google Scholar
  64. 64.
    N. Geodel, S. Schomann, and T. W. M. Clemens, GPU accelerated Adams-Bashforth multirate discontinuous Galerkin FEM simulation of high-frequency electromagnetic fields, IEEE Trans. Mag., 46 (2010), pp. 2735–8.Google Scholar
  65. 65.
    N. Goedel, N. N. T. Warburton, and M. Clemens, Scalability of higher-order discontinuous Galerkin FEM computations for solving electromagnetic wave propagation problems on GPU clusters, IEEE Trans. Mag., 46 (2010), pp. 3469–72.Google Scholar
  66. 66.
    J. Gopalakrishnan and J. Pasciak, Overlapping Schwarz preconditioners for indefinite time harmonic Maxwell equations, Math. Comput., 72 (2003), pp. 1–15.Google Scholar
  67. 67.
    V. Gradinaru and R. Hiptmair, Whitney elements on pyramids, ETNA, 8 (1999), pp. 154–68.Google Scholar
  68. 68.
    R. Graglia, D. Wilton, and A. Peterson, Higher order interpolatory vector bases for computational electromagnetics, IEEE Trans. Antennas Propagat., 45 (1997), pp. 329–342.Google Scholar
  69. 69.
    M. Grote and T. Mitkova, Explicit local time-stepping methods for Maxwell’s equations, J. Comput. Appl. Math., 234 (2010), pp. 3283–302.Google Scholar
  70. 70.
    M. Grote, A. Schneebeli, and D. Schötzau, Interior penalty discontinuous Galerkin method for Maxwell’s equations:Energy norm error estimates, IMA J. Numer. Anal., 204 (2007), pp. 375–86.Google Scholar
  71. 71.
    M. Grote, A. Schneebeli, and D. Schötzau, Interior penalty discontinuous Galerkin method for Maxwell’s equations: Optimal L 2 -norm error estimates, IMA J. Numer. Anal., 28 (2008), pp. 440–68.Google Scholar
  72. 72.
    M. Grote and D. Schoetzau, Optimal Error Estimates for the Fully Discrete Interior Penalty DG Method for the Wave Equation, JOURNAL OF SCIENTIFIC COMPUTING, 40 (2009), pp. 257–272.Google Scholar
  73. 73.
    T. Ha-Duong, On retarded potential boundary integral equations and their discretizations, in Topics in Computational Wave Propagation: Direct and Inverse Problems, M. Ainsworth, ed., Springer, 2003, pp. 301–36.Google Scholar
  74. 74.
    T. Ha-Duong, B. Ludwig, and I. Terrasse, A galerkin BEM for transient acoustic scattering by an absorbing obstacle, Int. J. Numer. Meth. Engng., 57 (2003), pp. 1845–82.Google Scholar
  75. 75.
    W. Hackbusch, W. Kress, and S. Sauter, Sparse convolution quadrature for time domain boundary integral formulations of the wave equation, IMA J. Numer. Anal., 29 (2009), pp. 158–79.Google Scholar
  76. 76.
    T. Hagstrom, Warburton, and D. Givoli, Radiation boundary conditions for time-dependent waves based on complete plane wave expansions, J. Comput. Appl. Math., 234 (2010), pp. 1988–95.Google Scholar
  77. 77.
    Y. Haugazeau and P. Lacoste, Condenstaion de la matrice masse pour les éléments finis mixtes de h(rot), Comptes Rendu, Series 1, 316 (1993), pp. 509–12.Google Scholar
  78. 78.
    Y. Heaugazeau and P. Lacoste, Lumping of the mass matrix for 1st-order mixed finite elements in h(curl), in Mathematical and Numerical Aspects of Wave Propagation, R. Kleinman, T. Angell, D. Colton, F. Santosa, and I. Stakgold, eds., SIAM, Philadelphia, 1993, pp. 259–68.Google Scholar
  79. 79.
    J. Hesthaven and T. Warburton, Nodal high-order methods on unstructured grids - I. Time-domain solution of Maxwell’s equations, J. Comput. Phys., 181 (2002), pp. 186–221.Google Scholar
  80. 80.
    R. Hiptmair, Multigrid method for Maxwell’s equations, SIAM J. Numer. Anal., 36 (1998), pp. 204–25.Google Scholar
  81. 81.
    R. Hiptmair, Discrete Hodge-operators: An algebraic perspective, Journal of Electromagnetic Waves and Applications, 15 (2001), pp. 343–4.Google Scholar
  82. 82.
    R. Hiptmair, Finite elements in computational electromagnetism, Acta Numerica, 11 (2002), pp. 237–339.Google Scholar
  83. 83.
    H. Holter, Some experiences from FDTD analysis of infinite and finite multi-octave phased arrays, IEEE Trans. Antennas Propagat., 50 (2002), pp. 1725–31.Google Scholar
  84. 84.
    W. Kress and S. Sauter, Numerical treatment of retarded boundary integral equations by sparse panel clustering, IMA J. Numer. Anal., 28 (2008), pp. 162–85.Google Scholar
  85. 85.
    T. Lahivaara and T. Huttunen, A non-uniform basis order for the discontinuous Galerkin method of the acoustic and elastic wave equations, Applied Numerical Mathematics, 61 (2011), pp. 473–86.Google Scholar
  86. 86.
    A. Laliena and F. Sayas, Theoretical aspects of the application of convolution quadrature to scattering of acoustic waves, Numer. Math., 112 (2009), pp. 637–78.Google Scholar
  87. 87.
    P. Lax, Functional Analysis, Wiley, New York, 2002.Google Scholar
  88. 88.
    M. Lazebnik, M. Okoniewski, J. Booske, and S. Hagness, Highly accurate Debye models for normal and malignant breast tissue dielectric properties at microwave frequencies, IEEE Microwave and Wireless Components Letters, 17 (2007), pp. 822–4.Google Scholar
  89. 89.
    J. Lee and B. Fornberg, Some unconditionally stable time stepping methods for the 3D Maxwell’s equations, J. Comput. Appl. Math., 166 (2004), pp. 497–523.Google Scholar
  90. 90.
    J.-F. Lee, WETD - A finite element time domain approach for solving Maxwell’s equations, IEEE Microwave and Guided Wave Lett., 4 (1994), pp. 11–3.Google Scholar
  91. 91.
    J.-F. Lee, R. Lee, and A. Cangellaris, Time-domain finite-element methods, IEEE Trans. Antennas Propagat., 45 (1997), pp. 430–42.Google Scholar
  92. 92.
    J. Li, Finite element study of the Lorentz model in metamaterials, Comput. Meth. Appl. Mech. Eng., 200 (2011), pp. 626–37.Google Scholar
  93. 93.
    J. Li, Y. Chen, and V. Elander, Mathematical and numerical study of wave propagation in negative-index materials, Comput. Meth. Appl. Mech. Eng., 197 (2008), pp. 45–8.Google Scholar
  94. 94.
    J. Li and Z. Zhang, Unified analysis of time domain mixed finite element methods for Maxwell’s equations in dispersive media, Journal of Computational Mathematics, 28 (2010), pp. 693–710.Google Scholar
  95. 95.
    C. Lubich, Convolution quadrature and discretized operational calculus. I and II, Numer. Math., 52 (1988), pp. 129–45 and 413–425.Google Scholar
  96. 96.
    C. Lubich, On the multistep time discretization of linear initial-boundary value problems and their boundary integral equations, Numer. Math., 67 (1994), pp. 365–89.Google Scholar
  97. 97.
    C. Lubich and A. Ostermann, Runge-Kutta methods for parabolic equations and convolution quadratur, Math. Comput., 60 (1993), pp. 105–31.Google Scholar
  98. 98.
    C. Makridakis and P. Monk, Time-discrete finite element schemes for Maxwell’s equations, RAIRO - Math. Model. Numer. Anal., 29 (1995), pp. 171–97.Google Scholar
  99. 99.
    W. McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, Cambridge, 2000.Google Scholar
  100. 100.
    T. W. M.Clemens, Transient eddy-current calculation with the FI-method, IEEE Trans. Mag., 35 (1999), pp. 1163–6.Google Scholar
  101. 101.
    J. Melenk and S. Sauter, Convergence analysis for finite element discretizations of the Helmholtz equation with Dirichlet-to-Neumann boundary conditions, Math. Comput., 79 (2010), pp. 1871–914.Google Scholar
  102. 102.
    P. Monk, Analysis of a finite element method for Maxwell’s equations, SIAM J. Numer. Anal., 29 (1992), pp. 714–29.Google Scholar
  103. 103.
    P. Monk, An analysis of Nédélec’s method for the spatial discretization of Maxwell’s equations, J. Comput. Appl. Math., 47 (1993), pp. 101–21.Google Scholar
  104. 104.
    P. Monk, Finite Element Methods for Maxwell’s Equations, Oxford University Press, Oxford, 2003.Google Scholar
  105. 105.
    P. Monk and A. Parrott, Phase-accuracy comparisons and improved far-field estimates for 3-D edge elements on tetrahedral meshes, J. Comput. Phys., 170 (2001), pp. 614–41.Google Scholar
  106. 106.
    J. Nédélec, Mixed finite elements in \({\mathbb{R}}^{3}\), Numer. Math., 35 (1980), pp. 315–41.Google Scholar
  107. 107.
    J. Nédélec, A new family of mixed finite elements in \({\mathbb{R}}^{3}\), Numer. Math., 50 (1986), pp. 57–81.Google Scholar
  108. 108.
    N.NIGAM and J. PHILLIPS, High-order finite elements on pyramids. accepted, IMA J. Numer.Anal., 2011.Google Scholar
  109. 109.
    J. Proakis and D. Manolakis, Digital Signal Processing: Principles, Algorithms, and Applications, Prentice Hall, Upper Saddle River, NJ, third edition ed., 1996.Google Scholar
  110. 110.
    S. Rao, D. Wilton, and A. Glisson, Electromagnetic scattering by surfaces of arbitrary shap, IEEE Trans. Antennas Propagat., 30 (1982), pp. 409–18.Google Scholar
  111. 111.
    P. Raviart and J. Thomas, Primal hybrid finite element methods for 2nd order elliptic equations, Math. Comput., 31 (1977), pp. 391–413.Google Scholar
  112. 112.
    R. Rieben, G. Rodrigue, and D. White, A high order mixed vector finite element method for solving the time dependent Maxwell equations on unstructured grids, J. Comput. Phys., 204 (2005), pp. 490–519.Google Scholar
  113. 113.
    D. Riley and J. Jin, Finite-element time-domain analysis of electrically and magnetically dispersive periodic structures, IEEE Trans. Antennas Propagat., (2008), pp. 3501–9.Google Scholar
  114. 114.
    T. Rylander and A. Bondeson, Stability of explicit-implicit hybrid time-stepping schemes for Maxwell’s equations, J. Comput. Phys., 179 (2002), pp. 426–438.Google Scholar
  115. 115.
    M. Schanz and H. Antes, A new visco- and elastodynamic time domain boundary element formulation, Computational Mechanics, 20 (1997), pp. 452–9.Google Scholar
  116. 116.
    S. Schomann, N. Goedel, T. Warburton, and M. Clemens, Local timestepping techniques using Taylor expansion for modeling electromagnetic wave propagation with discontinuous Galerkin-FEM, IEEE Trans. Mag., 46 (2010), pp. 3504–7.Google Scholar
  117. 117.
    W. Sha, X. Wu, Z. Huang, and M. Chen, Waveguide simulation using the high-order symplectic finite-difference time-domain scheme, Progress In Electromagnetics Research B, 13 (2009), pp. 217–56.Google Scholar
  118. 118.
    B. Shanker, A. Ergin, M. Lu, and E. Michielssen, Fast analysis of transient electromagnetic scattering phenomena using the multilevel plane wave time domain algorithm, IEEE Trans. Antennas Propagat., 51 (2003), pp. 628–41.Google Scholar
  119. 119.
    B. Shanker, A. A. Ergin, K. Aygun, and E. Michielssen, Analysis of transient electromagnetic scattering phenomena using a two-level plane wave time-domain algorithm, IEEE Trans. Antennas Propagat., 48 (2000), pp. 510–23.Google Scholar
  120. 120.
    S. Shaw, Finite element approximation of Maxwell’s equations with Debye memory. to appear in Advances in Numerical Analysis.Google Scholar
  121. 121.
    A. Taflove, Computational Electrodynamics, Artech House, Boston, 1995.Google Scholar
  122. 122.
    I. Terrasse, Résolution mathématique et numérique des équations de Maxwell instationnaires par une méthode de potentiels retardés, Spécialité: Mathématiques Appliquées, Ecole Polytechnique, Paris, France, 1993.Google Scholar
  123. 123.
    A. Toselli, Overlapping Schwarz methods for Maxwell’s equations in three dimensions, Numer. Math., 86 (2000), pp. 733–52.Google Scholar
  124. 124.
    X. Wang and D. Weile, Implicit Runge-Kutta methods for the discretization of time domain integral equations. IEEE Trans. Antennas Propagat., (2011), pp. 4651–63.Google Scholar
  125. 125.
    X. Wang, R. Wildman, D. Weile, and P. Monk, A finite difference delay modeling approach to the discretization of the time domain integral equations of electromagnetism, IEEE Trans. Antennas Propagat., 56 (2008), pp. 2442–52.Google Scholar
  126. 126.
    J. Webb and B. Forghani, Hierarchal scalar and vector tetrahedra, IEEE Trans. Mag., 29 (1993), pp. 1495–8.Google Scholar
  127. 127.
    T. Weiland, A discretization method for the solution of Maxwell’s equations for six-component fields, Electronics and Communications AEUE, 31 (1977), pp. 116–120.Google Scholar
  128. 128.
    T. Weiland, Numerical solution of Maxwell’s equation for static, resonant and transient problems, in Studies in Electrical and Electronic Engineering 28B, T. Berceli, ed., URSI International Symposium on Electromagnetic Theory Part B, Elsevier, New York, 1986, pp. 537–42.Google Scholar
  129. 129.
    D. Weile, G. Pisharody, N.-W. Chen, B. Shanker, and E. Michielssen, A novel scheme for the solution of the time-domain integral equations of electromagnetics, IEEE Trans. Antennas Propagat., 52 (2004), pp. 283–95.Google Scholar
  130. 130.
    Wikipedia: Finite-difference time-domain method., 2010.
  131. 131.
    R. Wildman and D. Weile, An accurate broad-band method of moments using higher order basis functions and tree-loop decomposition, IEEE Trans. Antennas Propagat., 52 (2004), pp. 3005–11.Google Scholar
  132. 132.
    K. Yee, Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media, IEEE Trans. Antennas Propagat., 16 (1966), pp. 302–7.Google Scholar
  133. 133.
    A. Yilmaz, D. W. E. Michielssen, and J.-M. Jin, A fast Fourier transform accelerated marching-on-in-time algorithm for electromagnetic analysis, Electromagnetics, 21 (2001), pp. 181–97.Google Scholar
  134. 134.
    A. Yilmaz, D. Weile, E. Michielssen, and J.-M. Jin, A hierarchical FFT algorithm (HIL-FFT) for the fast analysis of transient electromagnetic scattering phenomena, IEEE Trans. Antennas Propagat., (2002).Google Scholar
  135. 135.
    J. Zhao, Analysis of the finite element method for time-dependent Maxwell problems, Math. Comput., 247 (2003), pp. 1089–105.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  1. 1.Department of Mathematical SciencesUniversity of DelawareNewarkUSA

Personalised recommendations