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Multigrid methods for boundary integral equations. (English) Zbl 0543.65015

Multigrid methods are applied for solving algebraic systems of equations that occur to the numerical treatment of boundary integral equations of the first and second kind. These methods, originally formulated for partial differential equations of elliptic type, combine relaxation schemes and coarse grid corrections. The choice of the relaxation scheme is found to be essential to attain a fast convergent iterative process. Theoretical investigations show that the presented relaxation scheme provides a multigrid algorithm of which the rate of convergence increases with the dimension of the finest grid. This is illustrated for the calculation of potential flow around an aerofoil.

MSC:

65F10 Iterative numerical methods for linear systems
65R20 Numerical methods for integral equations
65N35 Spectral, collocation and related methods for boundary value problems involving PDEs
76B10 Jets and cavities, cavitation, free-streamline theory, water-entry problems, airfoil and hydrofoil theory, sloshing
35C15 Integral representations of solutions to PDEs
35J25 Boundary value problems for second-order elliptic equations
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References:

[1] Barnard, A.C.L., Duck, I.M., Lynn, M.S., Timlake, W.P.: The application of electromagnetic theory to electrocardiology, II. Numerical solution of the integral equation. Biophys. J.7, 463-491 (1967) · doi:10.1016/S0006-3495(67)86599-8
[2] Berkhoff, J.C.W.: Diffraction of water waves. In: Colloquium Numerical Treatment of Integral Equations, (H.J.J. te Riele, ed.), pp. 241-258. MC-Syllabus 41, Mathematisch Centrum, Amsterdam 1979 · Zbl 0428.76020
[3] Bland, S.R.: The two-dimensional oscillating airfoil in a wind tunnel in subsonic compressible flow. Ph. D. Thesis, University of North Carolina 1968 · Zbl 0198.29903
[4] Bueckner, H.F.: Field singularities and related integral representations. In: Mechanics of fracture, Vol. 1; Methods of Analysis and Solutions of Crack Problems, (G.C. Sih, ed.), pp. 239-314. Leyden: Noordhoff 1973 · Zbl 0319.73055
[5] Costabel, M., Stephan, E.: Boundary integral equations for mixed boundary values problems in polygonal domains and Galerkin approximation. Preprint Nr. 593, Technical University Darmstadt, Dept. Mathematics 1981 · Zbl 0655.65129
[6] Costabel, M., Stephan, E.: Curvature terms in the asymptotic expansions for solutions of boundary integral equations on curved polygons. Preprint Nr. 673, Technical University Darmstadt, Dept. Mathematics 1982 · Zbl 0538.35022
[7] Edwards, T.W., van Bladel, K.: Electrostatic dipole moment of a dielectric cube. Appl. Sci. Res.9, 151-155 (1961). · doi:10.1007/BF02930888
[8] Erdogan, F., Gupta, G.: The stress analysis of multi-layered composites with a flaw. Int. J. Solids Struct.7, 39-61 (1971) · Zbl 0205.55501 · doi:10.1016/0020-7683(71)90017-5
[9] Filippi, P.: Layer potentials and acoustic diffraction. J. Sound Vib.54, 473-500 (1977) · Zbl 0368.76073 · doi:10.1016/0022-460X(77)90607-1
[10] Fromme, J., Golberg, M.A.: Numerical solution of a class of integral equations arising in two-dimensional aerodynamics. J. Optimization Theory Appl.24, 169-206 (1978) · Zbl 0401.65078 · doi:10.1007/BF00933186
[11] Hackbusch, W.: Die schnelle Aufl?sung der Fredholmschen Integralgleichung zweiter Art. Beitr. Numer. Math.9, 47-62 (1981) · Zbl 0458.65108
[12] Hemker, P.W., Schippers, H.: Multiple grid methods for the solution of Fredholm integral equations of the second kind. Math. Comput.36, 215-232 (1981) · Zbl 0463.65086 · doi:10.1090/S0025-5718-1981-0595054-2
[13] Hess, J.L., Smith, A.M.O.: Calculation of potential flow about arbitrary bodies. Prog. Aero Sci.8, 1-138 (1967) · Zbl 0204.25602 · doi:10.1016/0376-0421(67)90003-6
[14] Hsiao, G.C., Maccamy, R.C.: Solution of boundary value problems by integral equations of the first kind. SIAM Rev.15, 687-705 (1973) · Zbl 0265.45009 · doi:10.1137/1015093
[15] Hsiao, G.C., Kopp, P., Wendland, W.L.: A Galerkin collocation method for some integral equations of the first kind. Computing25, 89-130 (1980) · Zbl 0431.65078 · doi:10.1007/BF02259638
[16] Jaswon, M.A., Symm, G.T.: Integral equation methods in potential theory and elastostatics. London: Academic Press 1977 · Zbl 0414.45001
[17] Jones, D.S.: Integral equations for the exterior acoustic problem. Q. J. Mech. Appl. Math.27, 129-142 (1974) · Zbl 0281.45006 · doi:10.1093/qjmam/27.1.129
[18] Martensen, E.: Berechnung der Druckverteilung an Gitterprofilen in ebener Potentialstr?mung mit einer Fredholmschen Integralgleichung. Arch. Ration. Mech. Anal.3, 235-270 (1959) · Zbl 0204.25603 · doi:10.1007/BF00284179
[19] Radon, J.: ?ber die Randwertaufgaben beim logarithmischen Potential. Sitz. ver. Akad. Wiss. Wien128, 1123-1167 (1919) · JFM 47.0457.01
[20] Rizzo, F.J.: An integral equation to boundary value problems of classical elastostatics. Q. Appl. Math.25, 83-95 (1967) · Zbl 0158.43406
[21] Schippers, H.: Multiple grid methods for equations of the second kind with applications in fluid mechanics. Ph. D. Thesis, Mathematisch Centrum. Amsterdam 1982, published as Mathematical Centre Tracts 163 · Zbl 0489.65074
[22] Schippers, H.: Application of multigrid methods for integral equations to two problems from fluid dynamics. J. Comput. Phys.48, 441-461 (1982) · Zbl 0505.76023 · doi:10.1016/0021-9991(82)90061-4
[23] Symm, G.T.: An integral equation method in conformal mapping. Numer. Math.9, 250-258 (1966) · Zbl 0156.16901 · doi:10.1007/BF02162088
[24] Wendland, W.L.: On Galerkin collocation methods for integral equations of elliptic boundary value problems. In: Numerical Treatment of Integral Equations. (J. Albrecht, L. Collatz, eds.), pp. 244-275. Intern. Ser. Numer. Math. 53. Basel: Birkh?user 1980
[25] Wendland, W.L.: I. Asymptotic convergence of boundary element methods. II. Integral equation methods for mixed boundary value problems. Preprint Nr. 611, Technical University Darmstadt, Dept. Mathematics 1981 · Zbl 0459.65077
[26] Wendland, W.L.: Boundary element methods and their asymptotic convergence. Lecture Notes of the CISM Summer-School on ?Theoretical Acoustics and Numerical Techniques?, (P. Filippi, ed.). International Centre for Mechanical Sciences. Udine, Italy 1982 (To appear in: Lecture Notes in Physics, Springer Verlag)
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