×

An over-relaxation method for the iterative solution of integral equations in scattering problems. (English) Zbl 0706.73078

Summary: A simple iterative method for solving many of the integral equations arising in scattering problems is presented. By introducing a relaxation parameter the equation is changed to one which may be solved as a Neumann series. An explicit choice of the relaxation parameter is proposed which does not require detailed knowledge of the spectrum nor does the method require the symmetrization of the, in general, non-selfadjoint integral operators that occur. Convergence of the method is demonstrated in examples where the Neumann series for the original equation either diverges or converges at a much lower rate.

MSC:

74S30 Other numerical methods in solid mechanics (MSC2010)
74J20 Wave scattering in solid mechanics
65R20 Numerical methods for integral equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Varga, R. S., Matrix Iterative Analysis (1962), Prentice-Hall: Prentice-Hall Englewood Cliffs, NJ · Zbl 0133.08602
[2] Patterson, W. M., Iterative Methods for the Solution of a Linear Operator Equation in Hilbert Space—A Survey, (Lecture Notes in Math., 394 (1974), Springer: Springer Berlin)
[3] van den Berg, P. M., Iterative computational techniques in scattering based upon the integrated square error criterion, IEEE Trans. Antennas Propagat., 32, 1063-1071 (1984)
[4] Sarkar, T. K., The conjugate gradient method as applied to electromagnetic field problems, IEEE-AP Newsletter, 28, 5-14 (1986)
[5] Bojarski, N. N., The \(k\)-space formulation of the scattering problem in the time domain, J. Acoust. Soc. Amer., 72, 570-584 (1982) · Zbl 0517.73025
[6] Ko, W. L.; Mittra, R., A new approach based on a combination of integral equation and asymptotic techniques for solving electromagnetic scattering problems, IEEE Trans. Antennas Propagat., 25, 187-197 (1977)
[7] Hellinger, E.; Toeplitz, O., Integralgleichungen und Gleichungen mit unendlichen Unbekannten, (Encyklopädie der Mathematischen Wissenschaften, IIc13 (1927), Teubner: Teubner Leipzig), 1335-1597 · JFM 53.0350.01
[8] Courant, R.; Hilbert, D., (Methods of Mathematical Physics, I (1953), Interscience: Interscience New York), 140-142
[9] Mullen, R. L.; Rencis, J. J., Iterative methods for solving boundary element equations, Computers and Structures, 25, 713-723 (1987) · Zbl 0603.73082
[10] Petryshyn, W. V., The generalized overrelaxation method for the approximate solution of operator equations in Hilbert space, J. Soc. Indust. Appl. Math., 10, 675-690 (1962) · Zbl 0106.31804
[11] Petryshyn, W. V., On a general iterative method for the approximate solution operator equations, Math. Comp., 17, 1-10 (1963) · Zbl 0111.31701
[12] Browder, F.; Petryshyn, W. V., The solution by iteration of linear functional equations in Banach spaces, Bull. Amer. Math. Soc., 72, 566-570 (1966) · Zbl 0138.08201
[13] Kammerer, W. J.; Nashed, M. Z., Iterative methods for best approximate solutions of linear integral equations of the first and second kinds, J. Math. Anal. Appl., 40, 547-573 (1972) · Zbl 0246.45015
[14] Landweber, L., An iteration formula for Fredholm integral equations of the first kind, Amer. J. Math., 73, 615-624 (1954) · Zbl 0043.10602
[15] Fridman, V., Method of successive approximations for a Fredholm integral equation of the first kind, Uspehi Mat. Nauk., 11, 233-234 (1956)
[16] Bialy, H., Iterative Bahandlung linearer Funktiongleichungen, Arch. Rational Mech. Anal., 4, 166-176 (1959) · Zbl 0154.40204
[17] Kleinman, R. E.; Roach, G. F., Iterative solutions of boundary integral equations in acoustics, (Proc. Roy. Soc. London A, 417 (1988)), 45-57 · Zbl 0647.65083
[18] Kleinman, R. E.; Roach, G. F.; Schuetz, L. S.; Shirron, J., An iterative solution to acoustic scattering by riged objects, J. Acoust. Soc. Amer., 84, 385-391 (1988)
[19] Chertock, G., Convergence of iterative solutions to integral equations for sound radiation, Q. Appl. Math., 26, 269-272 (1968) · Zbl 0182.22403
[20] Mur, G.; Nicia, A. J.A., Calculation of reflection and transmission coefficients in one-dimensional wave propagation problems, J. Appl. Phys., 47, 5218-5221 (1976)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.