Edge finite elements for the approximation of Maxwell resolvent operator

Daniele Boffi; Lucia Gastaldi

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2002)

  • Volume: 36, Issue: 2, page 293-305
  • ISSN: 0764-583X

Abstract

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In this paper we consider the Maxwell resolvent operator and its finite element approximation. In this framework it is natural the use of the edge element spaces and to impose the divergence constraint in a weak sense with the introduction of a Lagrange multiplier, following an idea by Kikuchi [14]. We shall review some of the known properties for edge element approximations and prove some new result. In particular we shall prove a uniform convergence in the L 2 norm for the sequence of discrete operators. These results, together with a general theory introduced by Brezzi, Rappaz and Raviart [8], allow an immediate proof of convergence for the finite element approximation of the time-harmonic Maxwell system.

How to cite

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Boffi, Daniele, and Gastaldi, Lucia. "Edge finite elements for the approximation of Maxwell resolvent operator." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 36.2 (2002): 293-305. <http://eudml.org/doc/245660>.

@article{Boffi2002,
abstract = {In this paper we consider the Maxwell resolvent operator and its finite element approximation. In this framework it is natural the use of the edge element spaces and to impose the divergence constraint in a weak sense with the introduction of a Lagrange multiplier, following an idea by Kikuchi [14]. We shall review some of the known properties for edge element approximations and prove some new result. In particular we shall prove a uniform convergence in the $L^2$ norm for the sequence of discrete operators. These results, together with a general theory introduced by Brezzi, Rappaz and Raviart [8], allow an immediate proof of convergence for the finite element approximation of the time-harmonic Maxwell system.},
author = {Boffi, Daniele, Gastaldi, Lucia},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {edge finite elements; time-harmonic Maxwell’s equations; mixed finite elements; time-harmonic Maxwell's equations, mixed finite elements; convergence},
language = {eng},
number = {2},
pages = {293-305},
publisher = {EDP-Sciences},
title = {Edge finite elements for the approximation of Maxwell resolvent operator},
url = {http://eudml.org/doc/245660},
volume = {36},
year = {2002},
}

TY - JOUR
AU - Boffi, Daniele
AU - Gastaldi, Lucia
TI - Edge finite elements for the approximation of Maxwell resolvent operator
JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 2002
PB - EDP-Sciences
VL - 36
IS - 2
SP - 293
EP - 305
AB - In this paper we consider the Maxwell resolvent operator and its finite element approximation. In this framework it is natural the use of the edge element spaces and to impose the divergence constraint in a weak sense with the introduction of a Lagrange multiplier, following an idea by Kikuchi [14]. We shall review some of the known properties for edge element approximations and prove some new result. In particular we shall prove a uniform convergence in the $L^2$ norm for the sequence of discrete operators. These results, together with a general theory introduced by Brezzi, Rappaz and Raviart [8], allow an immediate proof of convergence for the finite element approximation of the time-harmonic Maxwell system.
LA - eng
KW - edge finite elements; time-harmonic Maxwell’s equations; mixed finite elements; time-harmonic Maxwell's equations, mixed finite elements; convergence
UR - http://eudml.org/doc/245660
ER -

References

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  1. [1] C. Amrouche, C. Bernardi, M. Dauge and V. Girault, Vector potential in three-dimensional nonsmooth domains. Math Methods Appl. Sci. 21 (1998) 823–864. Zbl0914.35094
  2. [2] A. Bermúdez, R. Durán, A. Muschietti, R. Rodríguez and J. Solomin, Finite element vibration analysis of fluid-solid systems without spurious modes. SIAM J. Numer. Anal. 32 (1995) 1280–1295. Zbl0833.73050
  3. [3] D. Boffi, Fortin operator and discrete compactness for edge elements. Numer. Math. 87 (2000) 229–246. Zbl0967.65106
  4. [4] D. Boffi, A note on the de Rham complex and a discrete compactness property. Appl. Math. Lett. 14 (2001) 33–38. Zbl0983.65125
  5. [5] D. Boffi, F. Brezzi and L. Gastaldi, On the problem of spurious eigenvalues in the approximation of linear elliptic problems in mixed form. Math. Comp. 69 (2000) 121–140. Zbl0938.65126
  6. [6] D. Boffi, P. Fernandes, L. Gastaldi and I. Perugia, Computational models of electromagnetic resonators: analysis of edge element approximation. SIAM J. Numer. Anal. 36 (1998) 1264–1290. Zbl1025.78014
  7. [7] F. Brezzi and M. Fortin, Mixed and hybrid finite element methods. Springer-Verlag, New York (1991). Zbl0788.73002MR1115205
  8. [8] F. Brezzi, J. Rappaz and P.A. Raviart, Finite dimensional approximation of nonlinear problems. Part i: Branches of nonsingular solutions. Numer. Math. 36 (1980) 1–25. Zbl0488.65021
  9. [9] 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. Zbl1005.78012
  10. [10] L. Demkowicz, P. Monk, L. Vardapetyan and W. Rachowicz, de Rham diagram for h p finite element spaces. Comput. Math. Appl. 39 (2000) 29–38. Zbl0955.65084
  11. [11] L. Demkowicz and L. Vardapetyan, Modeling of electromagnetic absorption/scattering problems using h p -adaptive finite elements. Comput. Methods Appl. Mech. Engrg. 152 (1998) 103–124. Symposium on Advances in Computational Mechanics, Vol. 5 (Austin, TX, 1997). Zbl0994.78011
  12. [12] J. Descloux, N. Nassif and J. Rappaz, On spectral approximation. I. The problem of convergence. RAIRO Anal. Numér. 12 (1978) 97–112. Zbl0393.65024
  13. [13] P Fernandes and G. Gilardi, Magnetostatic and electrostatic problems in inhomogeneous anisotropic media with irregular boundary and mixed boundary conditions. Math. Models Methods Appl. Sci. 7 (1997) 957–991. Zbl0910.35123
  14. [14] F. Kikuchi, Mixed and penalty formulations for finite element analysis of an eigenvalue problem in electromagnetism. In Proceedings of the first world congress on computational mechanics (Austin, Tex., 1986), Vol. 64, pages 509–521, 1987. Zbl0644.65087
  15. [15] F. Kikuchi, On a discrete compactness property for the Nédélec finite elements. J. Fac. Sci., Univ. Tokyo, Sect. I A 36 (1989) 479–490. Zbl0698.65067
  16. [16] P. Monk, A finite element method for approximating the time-harmonic Maxwell equations. Numer. Math. 63 (1992) 243–261. Zbl0757.65126
  17. [17] P. Monk and L. Demkowicz, Discrete compactness and the approximation of Maxwell’s equations in ℝ 3 . Math. Comp. 70 (2001) 507–523. Zbl1035.65131
  18. [18] J.-C. Nédélec, Mixed finite elements in ℝ 3 . Numer. Math. 35 (1980) 315–341. Zbl0419.65069
  19. [19] J.-C. Nédélec, A new family of mixed finite elements in ℝ 3 . Numer. Math. 50 (1986) 57–81. Zbl0625.65107
  20. [20] J. Schöberl, Commuting quasi-interpolation operators for mixed finite elements. Preprint ISC-01-10-MATH, Texas A&M University, 2001. 
  21. [21] L. Vardapetyan and L. Demkowicz, h p -adaptive finite elements in electromagnetics. Comput. Methods Appl. Mech. Engrg. 169 (1999) 331–344. Zbl0956.78013

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