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Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.78.

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Consistency estimates for a double-layer potential and application to the numerical analysis of the boundary-element approximation of acoustic scattering by a penetrable object
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by A. Bendali and M. Souilah PDF
Math. Comp. 62 (1994), 65-91 Request permission

Abstract:

The consistency of the approximation of a double-layer potential when using a boundary-element method and approximating the curved boundary by a polygonal curve in two-dimensional problems or by a polyhedral surface in three-dimensional ones is investigated. The results are applied to the numerical analysis of the approximation of a model problem: the diffraction of a time-harmonic acoustic wave by a penetrable object.
References
  • S. Agmon, A. Douglis, and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. II, Comm. Pure Appl. Math. 17 (1964), 35–92. MR 162050, DOI 10.1002/cpa.3160170104
    • A. Bendali, Approximation par éléments finis de surface de problèmes de diffraction d’ondes électromagnétiques, Thèse de Doctorat d’Etat Es-Sciences Mathématiques, Univ. P. & M. Curie, Paris VI, Paris, 1984.
  • A. Bendali, Numerical analysis of the exterior boundary value problem for the time-harmonic Maxwell equations by a boundary finite element method. I. The continuous problem, Math. Comp. 43 (1984), no. 167, 29–46. MR 744923, DOI 10.1090/S0025-5718-1984-0744923-1
    • Y. Chang and R. F. Harrington, A surface formulation for characteristic modes of material bodies, Technical Report no. 2 AD/A 000 285 NTIS U.S. Department of Commerce, 1974.
  • Jacques Chazarain and Alain Piriou, Introduction to the theory of linear partial differential equations, Studies in Mathematics and its Applications, vol. 14, North-Holland Publishing Co., Amsterdam-New York, 1982. Translated from the French. MR 678605
    • D. Clair, Caractérisation du rayonnement plan E de cornets sectoraux plan H à l’aide d’une méthode bidimensionnelle, Thèse de Doctorat de $3^{\grave {\text {e}}\text {me}}$ cycle, Univ. P. & M. Curie, Paris VI, Paris, 1983.
  • 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
  • Martin Costabel, Boundary integral operators on Lipschitz domains: elementary results, SIAM J. Math. Anal. 19 (1988), no. 3, 613–626. MR 937473, DOI 10.1137/0519043
  • Martin Costabel and Ernst Stephan, A direct boundary integral equation method for transmission problems, J. Math. Anal. Appl. 106 (1985), no. 2, 367–413. MR 782799, DOI 10.1016/0022-247X(85)90118-0
  • M. Costabel and W. L. Wendland, Strong ellipticity of boundary integral operators, J. Reine Angew. Math. 372 (1986), 34–63. MR 863517, DOI 10.1515/crll.1986.372.34
    • N. Germain, Cours de mécanique des milieux continus, Masson, Paris, 1973. J. Giroire, Integral equation methods for exterior problem for Helmholtz equation, Rapport interne no. 40, Centre de Mathématiques Appliquées, Ecole Polytechnique, Palaiseau, France, 1978.
  • Jean Giroire, Integral equation methods for the Helmholtz equation, Integral Equations Operator Theory 5 (1982), no. 4, 506–517. MR 665145, DOI 10.1007/BF01694050
  • N. M. Günter, Potential theory and its applications to basic problems of mathematical physics, Frederick Ungar Publishing Co., New York, 1967. Translated from the Russian by John R. Schulenberger. MR 0222316
  • Mohamed Ali Hamdi, Une formulation variationnelle par équations intégrales pour la résolution de l’équation de Helmholtz avec des conditions aux limites mixtes, C. R. Acad. Sci. Paris Sér. II Méc. Phys. Chim. Sci. Univers Sci. Terre 292 (1981), no. 1, 17–20 (French, with English summary). MR 637242
    • R. F. Harrington, Characteristic modes for antennas and scatterers, Numerical and Asymptotic Techniques in Electromagnetics (R. Mittra, ed.), Topics in Applied Physics, vol. 3, Springer-Verlag, Berlin, Heidelberg, and New York, 1975, pp. 51-87.
  • George C. Hsiao and Wolfgang L. Wendland, A finite element method for some integral equations of the first kind, J. Math. Anal. Appl. 58 (1977), no. 3, 449–481. MR 461963, DOI 10.1016/0022-247X(77)90186-X
  • Claes Johnson and J.-Claude Nédélec, On the coupling of boundary integral and finite element methods, Math. Comp. 35 (1980), no. 152, 1063–1079. MR 583487, DOI 10.1090/S0025-5718-1980-0583487-9
  • R. E. Kleinman and P. A. Martin, On single integral equations for the transmission problem of acoustics, SIAM J. Appl. Math. 48 (1988), no. 2, 307–325. MR 933037, DOI 10.1137/0148016
  • R. Kreß and G. F. Roach, Transmission problems for the Helmholtz equation, J. Mathematical Phys. 19 (1978), no. 6, 1433–1437. MR 495653, DOI 10.1063/1.523808
    • G. Krishnasamy, F. Rizzo, and T. Rudolphi, Hypersingular boundary integral equations: Their occurrence, interpretation, regularization, and computation, Developments in Boundary Elements Methods, vol. 7: Adv. Dynamic Analysis (P. K. Banerjee and S. Kobayashi, eds.), Elsevier, Amsterdam, 1991, pp. 207-252.
  • M. N. Le Roux, Méthode d’éléments finis pour la résolution numérique de problèmes extérieurs en dimension $2$, RAIRO Anal. Numér. 11 (1977), no. 1, 27–60, 112 (French, with English summary). MR 448954, DOI 10.1051/m2an/1977110100271
  • J.-L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications. Vol. 1, Travaux et Recherches Mathématiques, No. 17, Dunod, Paris, 1968 (French). MR 0247243
    • J. C. Nédélec, Approximations des équations intégrales en mécanique et en physique, Cours EDF-CEA-IRIA, Centre de Mathématiques Appliquées, Ecole Polytechnique, Palaiseau, France, 1977.
  • 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
    • —, Integral equations with non-integrable kernels, Integral Equations Operator Theory 5 (1982), 561-572.
  • J.-C. Nédélec and J. Planchard, Une méthode variationnelle d’éléments finis pour la résolution numérique d’un problème extérieur dans $R^{3}$, Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge 7 (1973), no. R-3, 105–129 (French, with English summary). MR 424022
  • Umberto Neri, Singular integrals, Lecture Notes in Mathematics, Vol. 200, Springer-Verlag, Berlin-New York, 1971. Notes for a course given at the University of Maryland, College Park, Md., 1967. MR 0463818
  • Ralph S. Phillips, On the exterior problem for the reduced wave equation, Partial differential equations (Proc. Sympos. Pure Math., Vol. XXIII, Univ. California, Berkeley, Calif., 1971) Amer. Math. Soc., Providence, R.I., 1973, pp. 153–160. MR 0338545
  • Franz Rellich, Über das asymptotische Verhalten der Lösungen von $\Delta u+\lambda u=0$ in unendlichen Gebieten, Jber. Deutsch. Math.-Verein. 53 (1943), 57–65 (German). MR 17816
  • V. Rokhlin, Solution of acoustic scattering problems by means of second kind integral equations, Wave Motion 5 (1983), no. 3, 257–272. MR 708481, DOI 10.1016/0165-2125(83)90016-1
  • C. Schwab and W. L. Wendland, On numerical cubatures of singular surface integrals in boundary element methods, Numer. Math. 62 (1992), no. 3, 343–369. MR 1169009, DOI 10.1007/BF01396234
  • R. Seeley, Topics in pseudo-differential operators, Pseudo-Diff. Operators (C.I.M.E., Stresa, 1968) Edizioni Cremonese, Rome, 1969, pp. 167–305. MR 0259335
    • A. Sequeira, The coupling of boundary integral and finite element methods for the bidimensional steady Stokes problem, Rapport interne no. 82, Centre de Mathématiques Appliquées, Ecole Polytechnique, Palaiseau, France, 1982.
  • Gilbert Strang and George J. Fix, An analysis of the finite element method, Prentice-Hall Series in Automatic Computation, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1973. MR 0443377
    • A. Van Herk, Three dimensional analysis of magnetic fields in recording head configuration, IEEE Trans. Magnetics 5 MAG-16 (1980), 890-892.
  • Calvin H. Wilcox, Scattering theory for the d’Alembert equation in exterior domains, Lecture Notes in Mathematics, Vol. 442, Springer-Verlag, Berlin-New York, 1975. MR 0460927
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Additional Information
  • © Copyright 1994 American Mathematical Society
  • Journal: Math. Comp. 62 (1994), 65-91
  • MSC: Primary 65N38; Secondary 65R20, 76M15, 76Q05
  • DOI: https://doi.org/10.1090/S0025-5718-1994-1201067-9
  • MathSciNet review: 1201067