Resolution of the Maxwell equations in a domain with reentrant corners
Assous, F. ; Ciarlet, P. ; Sonnendrücker, E.
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 32 (1998), p. 359-389 / Harvested from Numdam
@article{M2AN_1998__32_3_359_0,
     author = {Assous, F. and Ciarlet, P. and Sonnendr\"ucker, E.},
     title = {Resolution of the Maxwell equations in a domain with reentrant corners},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     volume = {32},
     year = {1998},
     pages = {359-389},
     mrnumber = {1627135},
     zbl = {0924.65111},
     language = {en},
     url = {http://dml.mathdoc.fr/item/M2AN_1998__32_3_359_0}
}
Assous, F.; Ciarlet, P.; Sonnendrücker, E. Resolution of the Maxwell equations in a domain with reentrant corners. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 32 (1998) pp. 359-389. http://gdmltest.u-ga.fr/item/M2AN_1998__32_3_359_0/

[1] V.I. Agoshkov, Poincaré-Steklov's Operators and Domain Decomposition Methods in Finite Dimensional Spaces, in Glowinski, R. et al. eds, Domain Decomposition Methods for Partial Differential Equations, SIAM Philadelphia, (1988), 73-112. | MR 972513 | Zbl 0683.65097

[2] F. Assous, P. Jr. Ciarlet, J. Segré and E. Sonnendrücker. In préparation.

[3] F. Assous, P. Jr. Ciarlet and E. Sonnendrücker, Résolution des équations de Maxwell dans un domaine avec un coin rentrant. C. R. Acad. Sc. Paris Serie I 323 (1996) 203-208. | MR 1402544 | Zbl 0855.65131

[4] F. Assous, P. Jr. Ciarlet and E. Sonnendrücker, Résolution des équations de Maxwell dans un domaine 2D avec coins rentrants Partie I: Modélisation avec condition aux limites de type conducteur parfait, CEA, Technical Report, CEA-N-2813 (1996).

[5] F. Assous, P. Degond, E. Heintzé, P.-A. Raviart and J. Segré, On a Finite Element Method for Solving the Three-Dimensional Maxwell Equations, J. Comput. Phys. 109 (1993), 222-237. | MR 1253460 | Zbl 0795.65087

[6] I. Babuska, The finite element method with Lagrange multipliers, Numer. Math., 20 (1973), 179-192. | MR 359352 | Zbl 0258.65108

[7] I. Babuska, R. B. Kellog and Pitkäranta, Direct and inverse error estimates for finite elements with mesh refinements, Numer. Math., 33 (1979), 447-471. | MR 553353 | Zbl 0423.65057

[8] I. Babuska and H.-S. Oh, The p-Version of the Finite Element Method for Domains with Corners and for Infinite Domains, Numer. Methods Partial Differential Equations, 6 (1990), 371-392. | MR 1087251 | Zbl 0717.65084

[9] F. Brezzi, On the existence uniqueness and approximation of saddle point problems arising from Lagrange multipliers, RAIRO Anal. Numer. (1974), 129-151. | Numdam | MR 365287 | Zbl 0338.90047

[10] W. Cai, H. C Lee and H.-S. Oh, Coupling of Spectral Methods and the p-Version of the Finite Element Method for Ellitic Boundary Value Problems Containing Singularities, J. Comput. Phys. 108 (1993), 314-326. | MR 1242954 | Zbl 0790.65093

[11] M. Cessenat, Sur quelques opérateurs liés à l'équation de Helmholtz en coordonnées polaires, transformation H.K.L., C. R. Acad. Sci. Paris, Serie I 309 (1989), 25-30. | MR 1004933 | Zbl 0694.35040

[12] M. Cessenat, Résolution des problèmes de Helmholtz par séparation des variables en coordonnées polaires, C. R. Acad. Sci. Paris, Série I 309 (1989), 105-109. | MR 1004950 | Zbl 0688.35020

[13] P. Jr. Ciarlet, Tools for solving the div-curl problem with mixed boundary conditions in a polygonal domain. In preparation.

[14] P. Jr. Ciarlet and J. Zou, Finite Element Convergence for the Darwin Model to Maxwell's Equations, M2AN 7, 30(1996). | Numdam | Zbl 0887.65121

[15] M. Costabel, A Remark on the Regularity of Solutions of Maxwell's Equations on Lipschitz Domains, Math. Meth. Appl. Sci. 12, 2 (1990), 365-368. | MR 1048563 | Zbl 0699.35028

[16] M. Costabel, A Coercive Bilinear Form for Maxwell's Equations, J. Math. Anal. and Appl. 157, 2 (1991), 527-541. | MR 1112332 | Zbl 0738.35095

[17] M. Costabel and M. Dauge, Stable Asymptotics for Elliptic Systems on Plane Domains with Corners, Comm. PDE 19, 9 & 10 (1994), 1677-1726. | MR 1294475 | Zbl 0814.35024

[18] C. L. Cox and G. J. Fix, On the accuracy of least square methods in the presence of corner singularities, Comp. Math. Appl. 10, (1984), 463-471. | MR 783520 | Zbl 0573.65081

[19] M. Dauge, (1988), Elliptic boundary value problems on corner domains, Lecture Notes in Mathematics, 1341, Springer Verlag, Berlin. | MR 961439 | Zbl 0668.35001

[20] G. J. Fix (1986), Singular finite element method, in D. L. Doyer, M. Y. Hussami and R. G. Voigt (eds), Proc. ICASE Finite Element Theory and Application Workshop, Hampton, Virginia, Springer Verlag, Berlin, pp. 50-66. | MR 964480 | Zbl 0727.73070

[21] G. J. Fix, S. Gulati and G. I. Wakoff, On the use of singular function with finite element approximation, J Comp. Phys. 13, (1973), 209-228. | MR 356540 | Zbl 0273.35004

[22] P. Gérard and G. Lebeau, Diffusion d'une onde par un coin, J. Amer. Math. Soc. 6 (1993), 341-424. | MR 1157289 | Zbl 0779.35063

[23] V. Girault and P.-A. Raviart (1986), Finite Element Methods for Navier-Stokes Equations, Springer Series in Computational Mathematics, Springer Verlag, Berlin. | MR 851383 | Zbl 0585.65077

[24] D. Givoli, L. Rivkin and J. B. Keller, A Finite Element Method for Domains with Corners, Int. J. Numer. Methods Eng. 35 (1992), 1329-1345. | MR 1184244 | Zbl 0768.73072

[25] P. Grisvard (1985), Elliptic Problems in nonsmooth domains, Monographs and studies in Mathematics, 24, Pitman, London. | MR 775683 | Zbl 0695.35060

[26] P. Grisvard (1992), Singularities in boundary value problems, RMA 22, Masson, Paris. | MR 1173209 | Zbl 0766.35001

[27] T. J. R. Hughes and J. E. Akin, Techniques for developing "special" finite element shape functions with particular reference to singularities, Int. J. Numer. Methods Eng. 15 (1980), 733-751. | MR 580355 | Zbl 0428.73074

[28] J. B. Keller and D. Givoli, An Exact Non-Reflecting Boundary Condition, J. Comp. Phys. 82 (1988), 172-192. | MR 1005207 | Zbl 0671.65094

[29] O. Lafitte, The wave diffracted by a wedge with mixed boundary conditions, SIAM Annual Meeting, Stanford (1997). | MR 1635365 | Zbl 0963.78018

[30] O. Lafitte, Diffraction par une arête d'une onde électromagnétique normale à l'arête, in préparation.

[31] J.-L. Lions and E. Magenes (1968), Problèmes aux Limites Non Homogènes et Applications, Dunod, Paris. | Zbl 0165.10801

[32] M. Moussaoui, Espaces H(div, curl, Ω) dans un polygone plan. C. R. Acad. Sc. Paris, Série I 322 (1996) 225-229. | MR 1378257 | Zbl 0852.46034

[33] J. C. Nédélec, Mixed Finite Elements in R3, Numer. Math. 35 (1980), 315-341. | MR 592160 | Zbl 0419.65069

[34] D. Pathria and E. Karniadakis, Spectral Element Methods for Elliptic Problems in Nonsmooth Domains, J. Comput. Phys. 122 (1995), 83-95. | MR 1358523 | Zbl 0844.65082

[35] C. Weber, A local compactness theorem for Maxwell's equations, Math. Meth. Appl. Sci. 2 (1980), 12-25. | MR 561375 | Zbl 0432.35032