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

Christiansen, Snorre

doi:10.1090/S0025-5718-03-01581-3

Discrete Fredholm properties and convergence estimates for the electric field integral equation
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Math. Comp. 73 (2004), 143-167 Request permission

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

T. Abboud, Etude mathématique et numérique de quelques problèmes de diffraction d’ondes électromagnétiques, PhD thesis, Ecole Polytechnique, 1991.
Ivo Babuška, Error-bounds for finite element method, Numer. Math. 16 (1970/71), 322–333. MR 288971, DOI 10.1007/BF02165003
Ivo Babuška and A. K. Aziz, Survey lectures on the mathematical foundations of the finite element method, The mathematical foundations of the finite element method with applications to partial differential equations (Proc. Sympos., Univ. Maryland, Baltimore, Md., 1972) Academic Press, New York, 1972, pp. 1–359. With the collaboration of G. Fix and R. B. Kellogg. MR 0421106
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.
R. H. J. Germay, Généralisation de l’équation de Hesse, Ann. Soc. Sci. Bruxelles Sér. I 59 (1939), 139–144 (French). MR 86
Christine Bernardi, Claudio Canuto, and Yvon Maday, Generalized inf-sup conditions for Chebyshev spectral approximation of the Stokes problem, SIAM J. Numer. Anal. 25 (1988), no. 6, 1237–1271. MR 972452, DOI 10.1137/0725070
Daniele Boffi, Fortin operator and discrete compactness for edge elements, Numer. Math. 87 (2000), no. 2, 229–246. MR 1804657, DOI 10.1007/s002110000182
Daniele Boffi, Franco Brezzi, and Lucia Gastaldi, On the problem of spurious eigenvalues in the approximation of linear elliptic problems in mixed form, Math. Comp. 69 (2000), no. 229, 121–140. MR 1642801, DOI 10.1090/S0025-5718-99-01072-8
Armel de La Bourdonnaye, Décomposition de $H^{-1/2}_\textrm {div}(\Gamma )$ et nature de l’opérateur de Steklov-Poincaré du problème extérieur de l’électromagnétisme, C. R. Acad. Sci. Paris Sér. I Math. 316 (1993), no. 4, 369–372 (French, with English and French summaries). MR 1204306
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 French summary). MR 365287
Franco Brezzi, Jim Douglas Jr., and L. D. Marini, Two families of mixed finite elements for second order elliptic problems, Numer. Math. 47 (1985), no. 2, 217–235. MR 799685, DOI 10.1007/BF01389710
Franco Brezzi and Michel Fortin, Mixed and hybrid finite element methods, Springer Series in Computational Mathematics, vol. 15, Springer-Verlag, New York, 1991. MR 1115205, DOI 10.1007/978-1-4612-3172-1
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.
Michel Cessenat, Mathematical methods in electromagnetism, Series on Advances in Mathematics for Applied Sciences, vol. 41, World Scientific Publishing Co., Inc., River Edge, NJ, 1996. Linear theory and applications. MR 1409140, DOI 10.1142/2938
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.
Snorre Harald Christiansen and Jean-Claude Nédélec, Des préconditionneurs pour la résolution numérique des équations intégrales de frontière de l’électromagnétisme, C. R. Acad. Sci. Paris Sér. I Math. 331 (2000), no. 9, 733–738 (French, with English and French summaries). MR 1797761, DOI 10.1016/S0764-4442(00)01717-1
David L. Colton and Rainer Kress, Integral equation methods in scattering theory, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1983. A Wiley-Interscience Publication. MR 700400
S. Minakshi Sundaram, On non-linear partial differential equations of the hyperbolic type, Proc. Indian Acad. Sci., Sect. A. 9 (1939), 495–503. MR 0000089
L. Kantorovitch, The method of successive approximations for functional equations, Acta Math. 71 (1939), 63–97. MR 95, DOI 10.1007/BF02547750
L. Demkowicz, P. Monk, Ch. Schwab, and L. Vardapetyan, Maxwell eigenvalues and discrete compactness in two dimensions, Comput. Math. Appl. 40 (2000), no. 4-5, 589–605. MR 1772657, DOI 10.1016/S0898-1221(00)00182-6
George C. Hsiao and Ralph E. Kleinman, Mathematical foundations for error estimation in numerical solutions of integral equations in electromagnetics, IEEE Trans. Antennas and Propagation 45 (1997), no. 3, 316–328. MR 1437715, DOI 10.1109/8.558648
John David Jackson, Classical electrodynamics, 2nd ed., John Wiley & Sons, Inc., New York-London-Sydney, 1975. MR 0436782
Tosio Kato, Perturbation theory for linear operators, 2nd ed., Grundlehren der Mathematischen Wissenschaften, Band 132, Springer-Verlag, Berlin-New York, 1976. MR 0407617
Fumio Kikuchi, On a discrete compactness property for the Nédélec finite elements, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 36 (1989), no. 3, 479–490. MR 1039483
Josef Mall, Ein Satz über die Konvergenz von Kettenbrüchen, Math. Z. 45 (1939), 368–376 (German). MR 40, DOI 10.1007/BF01580290
J.-C. Nédélec, Curved finite element methods for the solution of singular integral equations on surfaces in $R^{3}$, Comput. Methods Appl. Mech. Engrg. 8 (1976), no. 1, 61–80. MR 455503, DOI 10.1016/0045-7825(76)90053-0
J.-C. Nédélec, Computation of eddy currents on a surface in $\textbf {R}^{3}$ by finite element methods, SIAM J. Numer. Anal. 15 (1978), no. 3, 580–594. MR 495761, DOI 10.1137/0715038
J.-C. Nédélec, Éléments finis mixtes incompressibles pour l’équation de Stokes dans $\textbf {R}^{3}$, Numer. Math. 39 (1982), no. 1, 97–112 (French, with English summary). MR 664539, DOI 10.1007/BF01399314
Jean-Claude Nédélec, Acoustic and electromagnetic equations, Applied Mathematical Sciences, vol. 144, Springer-Verlag, New York, 2001. Integral representations for harmonic problems. MR 1822275, DOI 10.1007/978-1-4757-4393-7
R. A. Nicolaides, Existence, uniqueness and approximation for generalized saddle point problems, SIAM J. Numer. Anal. 19 (1982), no. 2, 349–357. MR 650055, DOI 10.1137/0719021
Luc Paquet, Problèmes mixtes pour le système de Maxwell, Ann. Fac. Sci. Toulouse Math. (5) 4 (1982), no. 2, 103–141 (French, with English summary). MR 687546
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.
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) Lecture Notes in Math., Vol. 606, Springer, Berlin, 1977, pp. 292–315. MR 0483555
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.
Walter Rudin, Functional analysis, 2nd ed., International Series in Pure and Applied Mathematics, McGraw-Hill, Inc., New York, 1991. MR 1157815
Alfred H. Schatz, An observation concerning Ritz-Galerkin methods with indefinite bilinear forms, Math. Comp. 28 (1974), 959–962. MR 373326, DOI 10.1090/S0025-5718-1974-0373326-0
Gilbert Strang, Variational crimes in the finite element method, The mathematical foundations of the finite element method with applications to partial differential equations (Proc. Sympos., Univ. Maryland, Baltimore, Md., 1972) Academic Press, New York, 1972, pp. 689–710. MR 0413554
Nelson Dunford, A mean ergodic theorem, Duke Math. J. 5 (1939), 635–646. MR 98
I. Terrasse, Résolution mathématique et numérique des équations de Maxwell instationnaires par une méthode de potentiels retardés, PhD thesis, Ecole Polytechnique, 1993.
W. L. Wendland, Strongly elliptic boundary integral equations, The state of the art in numerical analysis (Birmingham, 1986) Inst. Math. Appl. Conf. Ser. New Ser., vol. 9, Oxford Univ. Press, New York, 1987, pp. 511–562. MR 921677