Matching of asymptotic expansions for waves propagation in media with thin slots II : the error estimates

Joly, Patrick; Tordeux, Sébastien

ESAIM: Modélisation mathématique et analyse numérique, Tome 42 (2008) no. 2, pp. 193-221.

We are concerned with a 2D time harmonic wave propagation problem in a medium including a thin slot whose thickness $\epsilon$ is small with respect to the wavelength. In a previous article, we derived formally an asymptotic expansion of the solution with respect to $\epsilon$ using the method of matched asymptotic expansions. We also proved the existence and uniqueness of the terms of the asymptotics. In this paper, we complete the mathematical justification of our work by deriving optimal error estimates between the exact solutions and truncated expansions at any order.

Classification : 35J05, 34E05, 78A45, 78A50
Mots clés : slit, slot, wave equation, Helmholtz equation, approximate model, matching of asymptotic expansions

[1] J. Beale, Scattering frequencies of reasonators. Comm. Pure Appl. Math. 26 (1973) 549-563. | MR | Zbl

[2] C. Butler and D. Wilton, General analysis of narrow strips and slots. IEEE Trans. Antennas Propag. 28 (1980) 42-48.

[3] G. Caloz, M. Costabel, M. Dauge and G. Vial, Asymptotic expansion of the solution of an interface problem in a polygonal domain with thin layer. Asymptotic Anal. 50 (2006) 121-173. | MR | Zbl

[4] M. Clausel, M. Duruflé, P. Joly and S. Tordeux, A mathematical analysis of the resonance of the finite thin slots. Appl. Numer. Math. 56 (2006) 1432-1449. | MR

[5] D. Crighton, A. Dowling, J.F. Williams, M. Heckl and F. Leppington, An asymptotic analysis, in Modern Methods in Analytical acoustics, Lecture Notes, Springer-Verlag, London (1992).

[6] J. Gilbert and R. Holland, Implementation of the thin-slot formalism in the finite-difference EMP code THREEDII. IEEE Trans. Nucl. Sci. 28 (1981) 4269-4274. | MR

[7] P. Harrington and D. Auckland, Electromagnetic transmission through narrow slots in thick conducting screens. IEEE Trans. Antennas Propag. 28 (1980) 616-622.

[8] A.M. Il'In, Matching of Asymptotic Expansions of Solutions of Boundary Value Problems, Translations of Mathematical Monographs 102. American Mathematical Society, Providence, RI (1992). Translated from the Russian by V. Minachin [V.V. Minakhin]. | MR | Zbl

[9] P. Joly and S. Tordeux, Asymptotic analysis of an approximate model for time harmonic waves in media with thin slots. ESAIM: M2AN 40 (2006) 63-97. | Numdam | MR

[10] P. Joly and S. Tordeux, Matching of asymptotic expansions for wave propagation in media with thin slots I: The asymptotic expansion. Multiscale Model. Simul. 5 (2006) 304-336. | MR | Zbl

[11] G. Kriegsmann, The flanged waveguide antenna: discrete reciprocity and conservation. Wave Motion 29 (1999) 81-95. | MR | Zbl

[12] N. Lebedev, Special functions and their applications. Revised English edition, translated and edited by Richard A. Silverman, Prentice-Hall Inc., Englewood Cliffs, N.J. (1965). | MR | Zbl

[13] M. Li, J. Nuebel, J. Drewniak, R. Dubroff, T. Hubing and T. Van Doren, EMI from cavity modes of shielding enclosures-fdtd modeling and measurements. IEEE Trans. Electromagn. Compat. 42 (2000) 29-38.

[14] V. Maz'Ya, S. Nazarov and B. Plamenevskii, Asymptotische Theorie elliptischer Randwertaufgaben in singulär gestörten Gebieten I, in Mathematische Monographien, Band 82, Akademie Verlag, Berlin (1991).

[15] V. Maz'Ya, S. Nazarov and B. Plamenevskii, Asymptotische Theorie elliptischer Randwertaufgaben in singulär gestörten Gebieten II, in Mathematische Monographien, Band 83, Akademie Verlag, Berlin (1991).

[16] B. Noble, Methods based on the Wiener-Hopf technique for the solution of partial differential equations, International Series of Monographs on Pure and Applied Mathematics 7. Pergamon Press, New York (1958). | MR | Zbl

[17] O. Oleinik, A. Shamaev and G. Yosifian, Mathematical Problems in Elasticity and Homogenization, Studies in Mathematics and Its Applications. North-Holland, Amsterdam (1992). | MR | Zbl

[18] S. Schot, Eighty years of Sommerfeld's radiation condition. Historia Math. 19 (1992) 385-401. | MR | Zbl

[19] A. Taflove, Computational Electrodynamics: The Finite-Difference Time-Domain Method. Artech House Inc., Boston, MA (1995). | MR | Zbl

[20] A. Taflove, K. Umashankar, B. Becker, F. Harfoush and K. Yee, Detailed fdtd analysis of electromagnetic fields penetrating narrow slots ans lapped joints in thick conducting screens. IEEE Trans. Antennas Propag. 36 (1988) 247-257.

[21] F. Tatout, Propagation d'une onde électromagnétique dans une fente mince. Propagation et réflexion d'ondes en élasticité. Application au contrôle. Ph.D. thesis, École normale supérieure de Cachan, France (1996).

[22] S. Tordeux, Méthodes asymptotiques pour la propagation des ondes dans les milieux comportant des fentes. Ph.D. thesis, Université de Versailles, France (2004).

[23] S. Tordeux and G. Vial, Matching of asymptotic expansions and multiscale expansion for the rounded corner problem. Tech. Rep. 2006-04, ETHZ, Seminar for applied mathematics (2006).

[24] M. Van Dyke, Perturbation methods in fluid mechanics. The Parabolic Press, Stanford, California (1975). | MR | Zbl

[25] G. Vial, Analyse multiéchelle et conditions aux limites approchées pour un problème de couche mince dans un domaine à coin. Ph.D. thesis, Université de Rennes I, France (2003).