Geodesic
Fast iteration method in the problem of waves interacting with a set of thin screens
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 53 (2013) no. 8, pp. 1374-1386 Cet article a éte moissonné depuis la source Math-Net.Ru

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A new iteration method is proposed for the wave equation describing the scattering of a harmonic wave from an arbitrary configuration in the form of an array of thin straight barriers. The problem is reduced to a system of boundary integral equations, which are discretized by applying the Belotserkovskii–Lifanov method. In discrete form, a finite number of systems with Toeplitz matrices (the number of systems is equal to the number of barriers) are solved at each iteration step by applying special fast methods. The algorithm is tested on several geometries, and its convergence in these cases is analyzed. 
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V. V. Popuzin; M. A. Sumbatyan; R. A. Tanyushin. Fast iteration method in the problem of waves interacting with a set of thin screens. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 53 (2013) no. 8, pp. 1374-1386. http://geodesic.mathdoc.fr/item/ZVMMF_2013_53_8_a12/

[1] Bowman J. J., Senior T. B. A., Uslenghi P. L. E., Electromagnetic and acoustic scattering by simple shapes, North Holland, Amsterdam, 1969

[2] Khenl Kh., Maue A., Vestpfal K., Teoriya difraktsii, Mir, M., 1964

[3] Shenderov E. L., Volnovye zadachi gidroakustiki, Sudostroenie, L., 1972

[4] Brebbiya K., Telles Zh., Vroubel L., Metody granichnykh elementov, Mir, M., 1987 | MR

[5] Sumbatyan M. A., Scalia A., Equations of mathematical diffraction theory, CRC Press, Boca Raton, Florida, 2005 | MR

[6] Scarpetta E., Sumbatyan M. A., “On the oblique wave penetration in elastic solids with a doubly periodic array of cracks”, Quart. Appl. Math., 58 (2000), 239–250 | MR | Zbl

[7] Linton C. M., McIver P., Handbook of mathematical techniques for wave/structure interactions, CRC Press, Boca Raton, Florida, 2001 | MR | Zbl

[8] Porter R., Evans D. V., “Wave scattering by periodic arrays of breakwaters”, Wave Motion, 23 (1996), 95–120 | DOI | MR | Zbl

[9] Lim R., Hackman R. H., “A formulation of multiple scattering by many bounded obstacles using a multicentered, T supermatrix”, J. Acoust. Soc. America, 91 (1992), 613–638 | DOI

[10] Voevodin V. V., Tyrtyshnikov E. E., Vychislitelnye protsessy s teplitsevymi matritsami, Nauka, M., 1987 | MR | Zbl

[11] Stewart M., “A superfast Toeplitz solver with improved numerical stability”, SIAM Journal on Matrix Analysis and Applications, 25:3 (2003), 669–693 | DOI | MR | Zbl

[12] Van Barel M., Heinig G., Kravanja P., “A stabilized superfast solver for nonsymmetric Toeplitz systems”, SIAM J. Matrix Analys. Appl., 23:2 (2001), 494–510 | DOI | MR | Zbl

[13] Chan T., “An optimal circulant preconditioner for Toeplitz”, SIAM J. Scientific and Statistical Comput., 74 (1986), 171–176

[14] Strang G., “A proposal for Toeplitz matrix calculations”, Studies in Appl. Math., 23:2 (2001), 494–510 | MR

[15] Tyrtyshnikov E. E., Metody chislennogo analiza, Akademiya, M., 2006 | Zbl

[16] Oseledets I. V., Tyrtyshnikov E. E., “A unifying approach to the construction of circulant preconditionets”, Linear Algebra and its Applicat., 418:2–3 (2006), 435–449 | DOI | MR | Zbl

[17] Lemb G., Gidrodinamika, Gostekhizdat, M.–L., 1947

[18] Vainikko G. M., Lifanov I. K., Poltavskii L. N., Chislennye metody v gipersingulyarnykh integralnykh uravneniyakh i ikh prilozheniya, Yanus-K, M., 2001

[19] Lifanov I. K., Metod singulyarnykh integralnykh uravnenii i chislennyi eksperiment, Yanus, M., 1995 | MR | Zbl

[20] Golub Dzh., Van-Loun Ch., Matrichnye vychisleniya, Mir, M., 1999