Hostname: page-component-78c5997874-fbnjt Total loading time: 0 Render date: 2024-11-05T19:37:27.882Z Has data issue: false hasContentIssue false
  • English
  • Français

Matching of asymptotic expansions for waves propagationin media with thin slots II: The error estimates

Published online by Cambridge University Press:  27 March 2008

Patrick Joly  and
Sébastien Tordeux
Patrick Joly
Affiliation:
Projet POEMS, Bâtiment 13, INRIA, Domaine de Voluceau - Rocquencourt - B.P. 105, 78153 Le Chesnay Cedex, France. [email protected]
Sébastien Tordeux
Affiliation:
Institut de Mathématiques de Toulouse, Université de Toulouse, 135 avenue de Rangueil, 31077 Toulouse Cedex 4, France. [email protected]
Get access

Abstract

We are concerned with a 2D time harmonic wave propagationproblem in a medium including a thin slot whose thickness εis small with respect to the wavelength. In a previous article, we derivedformally an asymptotic expansion of the solution with respect to εusing the method of matched asymptotic expansions. We also proved theexistence 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.

Keywords

Slitslotwave equationHelmholtz equationapproximate modelmatching of asymptotic expansions.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 42 , Issue 2 , March 2008 , pp. 193 - 221
Copyright
© EDP Sciences, SMAI, 2008

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Beale, J., Scattering frequencies of reasonators. Comm. Pure Appl. Math. 26 (1973) 549–563. CrossRef
Butler, C. and Wilton, D., General analysis of narrow strips and slots. IEEE Trans. Antennas Propag. 28 (1980) 42–48. CrossRef
Caloz, G., Costabel, M., Dauge, M. and Vial, G., Asymptotic expansion of the solution of an interface problem in a polygonal domain with thin layer. Asymptotic Anal. 50 (2006) 121–173.
Clausel, M., Duruflé, M., Joly, P. and Tordeux, S., A mathematical analysis of the resonance of the finite thin slots. Appl. Numer. Math. 56 (2006) 1432–1449. CrossRef
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).
Gilbert, J. and Holland, R., Implementation of the thin-slot formalism in the finite-difference EMP code THREEDII. IEEE Trans. Nucl. Sci. 28 (1981) 4269–4274. CrossRef
Harrington, P. and Auckland, D., Electromagnetic transmission through narrow slots in thick conducting screens. IEEE Trans. Antennas Propag. 28 (1980) 616–622. CrossRef
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].
Joly, P. and Tordeux, S., Asymptotic analysis of an approximate model for time harmonic waves in media with thin slots. ESAIM: M2AN 40 (2006) 63–97. CrossRef
Joly, P. and Tordeux, S., Matching of asymptotic expansions for wave propagation in media with thin slots I: The asymptotic expansion. Multiscale Model. Simul. 5 (2006) 304–336. CrossRef
Kriegsmann, G., The flanged waveguide antenna: discrete reciprocity and conservation. Wave Motion 29 (1999) 81–95. CrossRef
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).
Li, M., Nuebel, J., Drewniak, J., DuBroff, R., Hubing, T. and Van Doren, T., EMI from cavity modes of shielding enclosures-fdtd modeling and measurements. IEEE Trans. Electromagn. Compat. 42 (2000) 29–38.
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).
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).
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).
O. Oleinik, A. Shamaev and G. Yosifian, Mathematical Problems in Elasticity and Homogenization, Studies in Mathematics and Its Applications. North-Holland, Amsterdam (1992).
Schot, S., Eighty years of Sommerfeld's radiation condition. Historia Math. 19 (1992) 385–401. CrossRef
A. Taflove, Computational Electrodynamics: The Finite-Difference Time-Domain Method. Artech House Inc., Boston, MA (1995).
Taflove, A., Umashankar, K., Becker, B., Harfoush, F. and Yee, K., Detailed fdtd analysis of electromagnetic fields penetrating narrow slots ans lapped joints in thick conducting screens. IEEE Trans. Antennas Propag. 36 (1988) 247–257. CrossRef
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).
S. Tordeux, Méthodes asymptotiques pour la propagation des ondes dans les milieux comportant des fentes. Ph.D. thesis, Université de Versailles, France (2004).
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).
M. Van Dyke, Perturbation methods in fluid mechanics. The Parabolic Press, Stanford, California (1975).
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).