Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
   
Mobile Device Pairing
Green Open Access
Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

Numerical analysis of the exterior boundary value problem for the time-harmonic Maxwell equations by a boundary finite element method. II. The discrete problem


Author: A. Bendali
Journal: Math. Comp. 43 (1984), 47-68
MSC: Primary 65N30; Secondary 78-08, 78A45
MathSciNet review: 744924
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: With the help of curved and mixed finite elements, we introduce an approximate surface on which the discrete problem is defined and construct surface currents and charges which approximate the solution of the continuous problem studied in a previous part. We study the existence and uniqueness of the solution of the discrete problem and give estimates for the error between currents, charges, corresponding fields and their calculated approximations.


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

  • [1] Robert A. Adams, Sobolev spaces, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975. Pure and Applied Mathematics, Vol. 65. MR 0450957 (56 #9247)
  • [2] F. Brezzi, On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers, Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge 8 (1974), no. R-2, 129–151 (English, with loose French summary). MR 0365287 (51 #1540)
  • [3] Yvonne Choquet-Bruhat, Cecile DeWitt-Morette, and Margaret Dillard-Bleick, Analysis, manifolds and physics, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977. MR 0467779 (57 #7631)
  • [4] Philippe G. Ciarlet, The finite element method for elliptic problems, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978. Studies in Mathematics and its Applications, Vol. 4. MR 0520174 (58 #25001)
  • [5] G. J. Fix and R. A. Nicolaides, An analysis of mixed finite element approximations for periodic acoustic wave propagation, SIAM J. Numer. Anal. 17 (1980), no. 6, 779–786. MR 595443 (82f:76029), http://dx.doi.org/10.1137/0717065
  • [6] Tuong Ha Duong, A finite element method for the double-layer potential solutions of the Neumann exterior problem, Math. Methods Appl. Sci. 2 (1980), no. 2, 191–208. MR 570403 (81f:65094), http://dx.doi.org/10.1002/mma.1670020206
  • [7] R. F. Harrington, "Characteristic modes for antennas and scatterers," Numerical and Asymptotic Techniques in Electromagnetics (R. Mittra, ed.), Topics in Applied Physics, Vol. 3, Springer-Verlag, Berlin, Heidelberg, New York, 1975.
  • [8] G. C. Hsiao and W. L. Wendland, The Aubin-Nitsche lemma for integral equations, J. Integral Equations 3 (1981), no. 4, 299–315. MR 634453 (83j:45019)
  • [9] J. Giroire, Integral Equation Methods for Exterior Problems for the Helmholtz Equation, Rapport Interne no. 40, Centre de Mathématiques Appliquées, Ecole Polytechnique, Palaiseau, 1978.
  • [10] J.-L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications. Vol. 1, Travaux et Recherches Mathématiques, No. 17, Dunod, Paris, 1968 (French). MR 0247243 (40 #512)
  • [11] J. Nečas, Les Méthodes Directes en Théorie des Équations Elliptiques, Masson, Paris and Academia, Prague, 1967.
  • [12] J.-C. Nédélec, Curved finite element methods for the solution of singular integral equations on surfaces in 𝑅³, Comput. Methods Appl. Mech. Engrg. 8 (1976), no. 1, 61–80. MR 0455503 (56 #13741)
  • [13] J.-C. Nédélec, Computation of eddy currents on a surface in 𝑅³ by finite element methods, SIAM J. Numer. Anal. 15 (1978), no. 3, 580–594. MR 0495761 (58 #14409)
  • [14] D. T. Paris & F. K. Hurd, Basic Electromagnetic Theory, McGraw-Hill, Physical and Quantum Electronic Series, McGraw-Hill, New York, 1969.
  • [15] D. J. Poggio & E. K. Miller, "Solutions of three-dimensional scattering problems," Computer Techniques for Electromagnetics (R. Mittra, ed.), Pergamon Press, New York, 1973.
  • [16] S. S. M. Rao, D. R. Wilton & A. W. Glisson, "Electromagnetic scattering by surfaces of arbitrary shape," IEEE Trans. Antennas and Propagation, v. AP-30, 1982, pp. 409-418.
  • [17] P.-A. Raviart and J. M. Thomas, A mixed finite element method for 2nd order elliptic problems, Mathematical aspects of finite element methods (Proc. Conf., Consiglio Naz. delle Ricerche (C.N.R.), Rome, 1975) Springer, Berlin, 1977, pp. 292–315. Lecture Notes in Math., Vol. 606. MR 0483555 (58 #3547)
  • [18] Franz Rellich, Über das asymptotische Verhalten der Lösungen von Δ𝑢+𝜆𝑢=0 in unendlichen Gebieten, Jber. Deutsch. Math. Verein. 53 (1943), 57–65 (German). MR 0017816 (8,204c)
  • [19] J. M. Thomas, Sur l'Analyse Numérique des Méthodes d'Eléments Finis Hybrides et Mixtes, Thèse de Doctorat d'Etat, Univ. Paris VI, 1977.
  • [20] J. M. Thomas, Méthodes d'Éléments Finis Mixtes et Hybrides, Cours de D. E. A. 1980-1981, Univ. Paris VI, Laboratoire d'Analyse Numérique (LA 189), 1981.
  • [21] François Trèves, Introduction to pseudodifferential and Fourier integral operators. Vol. 2, Plenum Press, New York-London, 1980. Fourier integral operators; The University Series in Mathematics. MR 597145 (82i:58068)
  • [22] J. Van Bladel, Electromagnetic Fields, McGraw-Hill, New York, 1964.

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 65N30, 78-08, 78A45

Retrieve articles in all journals with MSC: 65N30, 78-08, 78A45


Additional Information

DOI: http://dx.doi.org/10.1090/S0025-5718-1984-0744924-3
PII: S 0025-5718(1984)0744924-3
Article copyright: © Copyright 1984 American Mathematical Society