Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 
 

 

Discrete Fredholm properties and convergence estimates for the electric field integral equation


Author: Snorre H. Christiansen
Journal: Math. Comp. 73 (2004), 143-167
MSC (2000): Primary 65N12, 65N38, 78M15
DOI: https://doi.org/10.1090/S0025-5718-03-01581-3
Published electronically: July 1, 2003
MathSciNet review: 2034114
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The Galerkin discretization of the Electric Field Integral Equation is reinvestigated. We prove quasi-optimal convergence estimates at nonresonant frequencies, using orthogonal splittings of the Galerkin space. At resonant frequencies we show that the spurious electric currents radiate only weakly in the exterior domain. This is achieved through the study of some finitely degenerated problems in terms of LBB Inf-Sup estimates and the use of discrete Helmholtz decompositions.


References [Enhancements On Off] (What's this?)

  • 1. T. Abboud, Etude mathématique et numérique de quelques problèmes de diffraction d'ondes électromagnétiques, PhD thesis, Ecole Polytechnique, 1991.
  • 2. I. Babuska, Error bounds for the finite element method, Numer. Math., Vol. 16, pp. 322-333, 1971. MR 44:6166
  • 3. I. Babuska, A.K. Aziz, Survey lectures on the mathematical foundations of the finite element method, pp. 3-359, in A.K. Aziz (ed.), ``The mathematical foundations of the finite element method with applications to partial differential equations'', Academic Press, New York, 1972. MR 54:9111
  • 4. I. Babuska, J. Osborn, Eigenvalue problems, pp. 641-788, in P.G. Ciarlet, J.-L. Lions (eds.), ``Handbook of numerical analysis, Vol. II, Finite element methods (Part 1)'', North-Holland, 1991.
  • 5. A. Bendali, Numerical analysis of the exterior boundary value problem for the time-harmonic Maxwell equations by a boundary finite element method, Part 1: The continuous problem, Math. Comp., Vol. 43, No. 167, pp. 29-46, 1984, Part 2: The discrete problem, Math. Comp., Vol. 43, No. 167, pp. 47-68, 1984. MR 86i:65071a, MR 86i:65071b
  • 6. C. Bernardi, C. Canuto, Y. Maday, Generalized Inf-Sup conditions for the Chebyshev spectral approximation of the Stokes problem, SIAM J. Numer. Anal., Vol. 25, No. 6, pp. 1237-1271, 1988. MR 90e:65151
  • 7. D. Boffi, Discrete compactness and Fortin operator for edge elements, Numer. Math., Vol. 87, No. 2, pp. 229-246, 2000. MR 2001k:65168
  • 8. D. Boffi, F. Brezzi, L. Gastaldi, On the problem of spurious eigenvalues in the approximation of linear elliptic problems in mixed form, Math. Comp., Vol. 69, No. 229, pp. 121-140, 1999. MR 2000i:65175
  • 9. A. DeLaBourdonnaye, Décomposition de $\mathrm{H}_{\operatorname{div}} ^{-1/2}(\Gamma)$ et nature de l'opérateur de Steklov-Poincaré du problème extérieur de l'électromagnétisme, C. R. Acad. Sci. Paris Sér. I Math., Vol. 316, No. 4, pp. 369-372, 1993. MR 93k:78004
  • 10. F. Brezzi, On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers, RAIRO Anal. Numér., Vol. 8, No. R-2, pp. 129-151, 1974. MR 51:1540
  • 11. F. Brezzi, J. Douglas (Jr.), L.D. Marini, Two families of mixed finite elements for second order elliptic problems, Numer. Math., Vol. 47, pp. 217-235, 1985. MR 87g:65133
  • 12. F. Brezzi, M. Fortin, Mixed and hybrid finite element methods, Springer-Verlag, 1991. MR 92d:65187
  • 13. A. Buffa, M. Costabel, C. Schwab, Boundary element methods for Maxwell's equations on non-smooth domains, Numer. Math., Vol. 92, No. 4, pp. 679-710, 2002.
  • 14. M. Cessenat, Mathematical methods in electromagnetism, Linear theory and applications, World Scientific Publishing Co., River Edge, NJ, 1996. MR 97j:78001
  • 15. S.H. Christiansen, F. Béreux, J.-C. Nédélec, J.-P. Martinaud, Algorithme de simulation électromagnétique, notamment des performances d'une antenne, Patent by Thomson-CSF Detexis, Reg. No. 0007456 at INPI, dated June 9, 2000.
  • 16. S.H. Christiansen, J.-C. Nédélec, Des préconditionneurs pour la résolution numérique des équations intégrales de frontière de l'électromagnétisme, C. R. Acad. Sci. Paris, Sér. I Math., Vol. 331, No. 9, pp. 733-738, 2000. MR 2001i:78022
  • 17. D.L. Colton, R. Kress, Integral equation methods in scattering theory, John Wiley & Sons, New York, 1983. MR 85d:35001
  • 18. M. Costabel, Boundary integral operators on Lipschitz domains: elementary results, SIAM J. Math. Anal., Vol. 19, No. 3, pp. 613-626, 1988. MR 89h:35090 10pt
  • 19. L. Demkowicz, Asymptotic convergence in finite and boundary element methods, Part 1, Theoretical results, Comput. Math. Appl., Vol. 27, No. 12, pp. 69-84, 1994, Part 2, The LBB constant for rigid and elastic scattering problems, Comput. Math. Appl., Vol. 28, No. 6, pp. 93-109, 1994. MR 95h:65080, MR 95k:65106
  • 20. L. Demkowicz, P. Monk, C. Schwab, L. Vardapetyan, Maxwell eigenvalues and discrete compactness in two dimensions, Comput. Math. Appl., Vol. 40, No. 4-5, pp. 589-605, 2000. MR 2002e:65166
  • 21. G.C. Hsiao, R.E. Kleinman, Mathematical foundations of error estimates in numerical solutions of integral equations in electromagnetics, IEEE Trans. Ant. Prop., Vol. 47, No. 3, pp. 316-328, 1997. MR 98a:78001
  • 22. J.D. Jackson, Classical electrodynamics, Second edition, John Wiley & Sons, New York-London-Sydney, 1975. MR 55:9721
  • 23. T. Kato, Perturbation theory for linear operators, Second edition, Springer-Verlag, Berlin-New York 1976. MR 53:11389
  • 24. F. Kikuchi, On a discrete compactness property for the Nédélec finite elements, J. Fac. Sci. Univ. Tokyo, Sect. 1A Math., Vol. 36, pp. 479-490, 1989. MR 91h:65173
  • 25. J.-L. Lions, E. Magenes, Problèmes aux limites non homogènes et applications, Vol. I, Vol. II, Dunod, Paris, 1968. MR 40:512, MR 40:513
  • 26. J.-C. Nédélec, Curved finite element methods for the solution of singular integral equations on surfaces in $\mathbb{R}^3$, Comput. Methods Appl. Mech. Engrg., Vol. 8, No. 1, pp. 61-80, 1976. MR 56:13741
  • 27. J.-C. Nédélec, Computation of eddy currents on a surface in $\mathbb{R}^3$ by finite element methods, SIAM J. Numer. Anal., Vol. 15, No. 3, pp. 580-594, 1978. MR 58:14409
  • 28. J.-C. Nédélec, Eléments finis mixtes incompressibles pour l'équation de Stokes dans $\mathbb{R}^3$, Numer. Math., Vol. 39, pp. 97-112, 1982. MR 83g:65111
  • 29. J.-C. Nédélec, Acoustic and Electromagnetic Equations, Integral Representations for Harmonic Problems, Springer-Verlag, 2001. MR 2002c:35003
  • 30. R.A. Nicolaides, Existence, uniqueness and approximation for generalized saddle point problems, SIAM J. Numer. Anal., Vol. 19, No. 2, pp. 349-357, 1982. MR 83d:49023
  • 31. L. Paquet, Problèmes mixtes pour le système de Maxwell, Ann. Fac. Sci. Toulouse Math., Vol. 4, No. 2, pp. 103-141, 1982. MR 84e:58075
  • 32. S.S.M. Rao, D.R. Wilton, A.W. Glisson, Electromagnetic scattering by surfaces of arbitrary shape, IEEE Trans. Ant. Prop. AP-30, pp. 409-418, 1982.
  • 33. P.A. Raviart, J.-M. Thomas, A mixed finite element method for $2$nd order elliptic problems, pp. 292-315, in I. Galligani, E. Magenes (eds.), ``Mathematical aspects of the finite element method'', Lecture Notes in Math., Vol. 606, Springer-Verlag, Berlin and New York, 1977. MR 58:3547
  • 34. J.E. Roberts, J.-M. Thomas, Mixed and hybrid methods, pp. 523-640, in P.G. Ciarlet, J.-L. Lions (eds.), ``Handbook of numerical analysis, Vol. II, Finite element methods (Part 1)'', North-Holland, 1991.
  • 35. W. Rudin, Functional analysis, Second edition, McGraw-Hill, New York, 1991. MR 92k:46001
  • 36. A.H. Schatz, An observation concerning Ritz-Galerkin methods with indefinite bilinear forms, Math. Comp., Vol. 28, pp. 959-962, 1974. MR 51:9526
  • 37. G. Strang, Variational crimes in the finite element method, pp. 689-710, in A.K. Aziz (ed.), ``The mathematical foundations of the finite element method with applications to partial differential equations'', Academic Press, New York, 1972. MR 54:1668
  • 38. M. Taylor, Partial differential equations, Vol. I Basic theory, Vol. II Qualitative studies in linear equations, Springer-Verlag, New York, 1996. MR 98b:35002b, MR 98b:35003
  • 39. I. Terrasse, Résolution mathématique et numérique des équations de Maxwell instationnaires par une méthode de potentiels retardés, PhD thesis, Ecole Polytechnique, 1993.
  • 40. W.L. Wendland, Strongly elliptic boundary integral equations, pp. 511-561, in A. Iserles, M. Powell (eds.), ``The state of the art in numerical analysis (Birmingham 1986)'', Inst. Math. Appl. Conf. Ser. New Ser. 9, Oxford Univ. Press, New York, 1987. MR 88m:65209

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2000): 65N12, 65N38, 78M15

Retrieve articles in all journals with MSC (2000): 65N12, 65N38, 78M15


Additional Information

Snorre H. Christiansen
Affiliation: Matematisk Institutt, P.B. 1053 Blindern, N-0316 Oslo, Norway
Email: snorrec@math.uio.no

DOI: https://doi.org/10.1090/S0025-5718-03-01581-3
Received by editor(s): December 26, 2000
Received by editor(s) in revised form: April 10, 2002
Published electronically: July 1, 2003
Additional Notes: This work received financial support from Thales Airborne Systems
Article copyright: © Copyright 2003 American Mathematical Society

American Mathematical Society