Summary
The slow viscous flow past a spatial body with corners and edges is investigated mathematically and numerically by means of a boundary element method. For the resulting algebraic system a multigrid solver is designed and analyzed. Due to an improved bound on the rate of convergence it proves to be preferable to that introduced earlier for related problems. A numerical example illustrates some of the proposed methods.
Similar content being viewed by others
References
Borchers, W.: Eine Fourier-Spektralmethode für das Stokes-Resolventenproblem. Numer. Math. (in press)
Borchers, W., Hebeker, F.K.: The boundary element spectral method and applications in 3-D viscous hydrodynamics. In: (C.A. Brebbia, M. Tanake, eds.), pp. 823–828 Boundary Elements VIII, Vol. 2. Berlin, Heidelberg, New York, Tokyo: Springer 1986
Brakhage, H., Werner, P.: Über das Dirichlet'sche Außenraumproblem für die Helmholtz'sche Schwingungsgleichung. Arch. Math.16, 325–329 (1965)
Brebbia, C.A., Maier, G. (eds.) Boundary Elements VII, Vol. 2. Berlin, Heidelberg, New York, Tokyo: Springer 1985
Bruhn, G., Wendland, W.L.: Über die näherungsweise Lösung von linearen Funktionalgleichungen. In: (L. Collatz, G. Meinardus, H. Unger, eds.), pp. 136–164. Funktionalalysis, Approximationstheorie, Numerische Mathematik. Basel: Birkhäuser 1967
Burago, Y.D., Maz'ja, V.G., Sapozhnikova, V.D.: On the theory of simple and double-layer potentials for domains with irregular boundaries. In: Problems in Mathematical Analysis (V.I. Smirnov, ed.), pp. 1–30. Vol. 1. New York: Consultants Bureau 1968
Costabel, M.: Boundary integral operators on curved polygons. Ann. Mat. Pura Appl., Ser. 4a133, 305–326 (1983)
Costabel, M.: Boundary integral operators on Lipschitz domains: elementary results. University of Darmstadt. Preprint 1985
Costabel, M., Stephan, E.: The normal derivative of the double layer potential on polygons and Galerkin approximation. Appl. Anat.16, 205–228 (1983)
Costabel, M., Stephan, E., Wendland, W.L.: On boundary integral equations of the first kind for the bi-Laplacean in a polygonal plane domain. Ann. Scuola Nor. Sup. Pisa, Ser. 4,10, 197–241 (1983)
Courant, R., Hilbert, D.: Methods of Mathematical Physics, Vol. 2. New York: Interscience 1953
Dunford, N., Schwartz, J.: Linear Operators, Vol. 1, New York: Wiley 1957
Eskin, G.I.: Boundary Value Problems for Elliptic Pseudoifferential Equations. Am. Math. Soc., Providence, RI, 1980
Fischer, T.M., Rosenberger, R.: A boundary integral method for the numerical computation of the forces exerted on a sphere in viscous incompressible flows near a plane wall. J. Appl. Math. Phys.38, 339–365 (1987)
Günter, N.M.: Die Potentialtheorie und ihre Anwendungen auf Grundaufgaben der mathematischen Physik. Leipzig: Teubner 1957
Grisvard, P.: Elliptic Problems in Nonsmooth Domains. London: Pitman 1985
Hackbusch, W.: Multigrid Methods. Berlin, Heidelberg, New York, Tokyo: Springer 1985
Hartmann, F.: Elastic potentials on piecewise smooth surfaces. J. Elasticity12, 31–50 (1982)
Hartmann, F.: The Somigliana identity on piecewise smooth surfaces. J. Elasticity11, 403–423 (1981)
Hebeker, F.K.: A theorem of Faxén and the boundary integral method for 3-D viscous incompressible flows. Techn. Rep., University of Paderborn 1982
Hebeker, F.K.: A boundary integral approach to compute the 3-D Oseen flow past a moving body. In: Numerical Methods in Fluid Dynamics (M. Pandolfi, R. Piva eds.), pp. 124–130. Braunschweig: Vieweg 1984
Hebeker, F.K.: On a multigrid method for the integral equations of 3-D Stokes flow. In: Efficient Solution of Elliptic Systems (W. Hackbusch ed.), pp. 67–73. Brauschweig: Vieweg 1984
Hebeker, F.K.: A boundary element method for Stokes equations in 3-D exterior domains. In: The Mathematics of Finite Elements and Applications (J.R. Whiteman, ed.), pp. 257–263, Vol. 5. London: Academic Press 1985
Hebeker, F.K.: Efficient boundary element methods for 3-D exterior viscous flows. Numer. Meth. PDE,2, 273–297 (1986)
Hebeker, F.K.: Efficient boundary element methods for 3-D viscous flows. In: Boundary Elements VII (C.A. Brebbia, G. Maier, eds.), pp. 9:37–44, Vol. 2. Berlin, Heidelberg, New York, Tokyo: Springer 1985
Hebeker, F.K.: On a new boundary element spectral method. In: Innovative Numerical Methods in Engineering (R.P. Shaw ed.), pp. 311–316. (Conf. at Atlanta, March 1986). Berlin, Heidelberg, New York, Tokyo: Springer 1986
Hess, J.L., Smith, A.M.O.: Calculation of potential flow about arbitrary bodies. In: Progress in Aeronautical Sciences8 (D. Kuchemann, ed.), pp. 1–138, Pergamon: New York 1967
Hsiao, G.C.: Integral representations of solutions for 2-D viscous flow problems. Integral Equations Oper. Theory5, 533–547 (1982)
Hsiao, G.C., Kress, R.: On an integral equation for the 2-D exterior Stokes problem. Appl. Numer. Math.1, 77–93 (1985)
Hsiao, G.C., MacCamy, R.C.: Singular perturbations for the 2-D viscous flow problem. Springer Lect. Notes Math.942, 229–244 (1982)
Hsiao, G.C., Stephan, E., Wendland, W.L.: On the integral equation method for the plane mixed boundary value problem with the Laplacean. Math. Methods Appl. Sci.1, 265–321 (1979)
Fischer, T.M., Hsiao, G.C., Wendland, W.L.: Singular perturbations for the exterior three-dimensional slow viscous flow problem. J. Math. Anal. Appl.110, 583–603 (1985)
Jörgens, K.: Lineare Integraloperatoren. Stuttgart: Teubner 1970
Kersten, H.: Ein neuer Zugang zum ersten Randwertproblem der Schwingungsgleichung in Gebieten mit nicht-glattem Rand. Habilitation Thesis, University of Aachen 1983
Kleinman, R.E., Wendland, W.L.: On Neumann's method for the exterior Neumann problem for the Helmholtz equation. J. Math. Anal. Appl.57, 170–202 (1977)
Kondratiev, V.A.: Boundary problems for elliptic equations in domains with conical or angular points. Trans. Mosc. Math. Soc.16, 227–313 (1967)
Kral, J.: Integral Operators in Potential Theory. Berlin, Heidelberg, New York, Tokyo: Springer 1980
Kupradze, V.D., Gegelia, T.G., Basheishvili, M.O., Burchuladze, T.V.: 3-D Problems of the Mathematical Theory of Elasticity and Thermoelasticity. Amsterdam: North Holland 1979
Ladyzhenskaja, O.A.: The Mathematical Theory of Viscous Incompressible Flow. New York: Gordon and Breach 1969
Leis, R.: Zur Eindeutigkeit der Randwertaufgaben der Helmholtz'schen Schwingungsgleichung. Math. Z.85, 141–159 (1964)
Novak, Z.: Use of the multigrid method for Laplacean problems in 3-D. In: Multigrid Methods (W. Hackbusch, U. Trottenberg, eds.), pp. 576–598. Berlin, Heidelberg, New York, Tokyo: Springer 1982
Panich, O.I.: On the question of the solvability of the exterior boundary value problem for the wave equation and Maxwell's equation. Russ. Math. Surv.20, 221–226 (1965)
Radon, J.: Über lineare Funktionaltransformationen und Funktionalgleichungen. Akad. Wiss., Math.-Naturw. Kl., Sitz.-ber. Abt. IIA,128, 1083–1121 (1919)
Radon, J.: Über die Randwertaufgaben beim logorithmischen Potential. Akad. Wiss., Math.-Naturw. K., Sitz.-ber. Abt. IIA,128, 1123–1167 (1919)
Schippers, H.: Multigrid methods for boundary integral equations. Numer. Math.46, 351–363 (1985)
Solonnikov, V.A.: On the Stokes equations in domains with nonsmooth boundaries, and on viscous incompressible flow with a free surface. In: Nonlienar Partial Differential Equations and their Applications (H. Brezis, J.L. Lions, eds.), pp. 340–423, Vol. 3. Boston: Pitman 1982
Stephan, E.: Boundary integral equations for mixed boundary value problems, screen and transmission problems in ℝ3. pp. 136, Habilitation Thesis, University of Darmstadt 1984
Stephan, E., Wendland, W.L.: An augmented Galerkin procedure for the boundary integral method applied to 2-D screen and crack problems. Appl. Anal.18, 183–220 (1984)
Varnhorn, W.: Zur Numerik der Gleichungen von Navier Stokes. PhD Thesis, University of Paderborn 1985
Wendland, W.L.: Die Behandlung von Randwertaufgaben im ℝ3 mit Hilfe von Einfach-und Doppelschichtpotentialen. Numer. Math.11, 380–404 (1968)
Wendland, W.L.: Boundary element methods and their asymptotic convergence. In: Theoretical Acoustics and Numerical Techniques (P. Filippi, ed.), pp. 135–216. CISM Courses and Lectures 277, Wien 1983
Wendland, W.L.: Bemerkungen zu Randelementmethoden bei Rissproblemen. Mathematica (Festschrift E. Mohr), Technical University of Berlin 1985, pp. 307–322
Wendland, W.L.: On some mathematical aspects of boundary element methods for elliptic problems. In: The Mathematics of Finite Elements and Applications (J.R. Whiteman, ed.), pp. 193–227. London: Academic Press 1985
Zhu, J.: A boundary integral equation method for the stationary Stokes problem in 3-D. In: Boundary Element Methods (C.A. Brebbia ed.), pp. 283–292. Berlin, Heidelberg, New York, Tokyo: Springer 1983
Leis, R.: Über die Dirichletische Randwertaufgabe zur Schwingungsgleichung. Z. Angew. Math. Mech.44, T45–47 (1964)
Maz'ja, V.G., Plamenevski, B.A.: Die erste Randwertaufgabe für klassische Gleichungen der mathematischen Physik in Gebieten mit stükweise glatten Rändern (Russian). I: Z. Anal. Anwend.2, 335–359 (1983). II: Z. Anal. Anwend.2, 523–551 (1983)
Dauge, M.: Stationary Stokes and Navier Stokes Systems on two-or three dimensional domains with corners. Part I: Linearized equaitons. Preprint, University of Nantes 1987
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Hebeker, F.K. On the numerical treatment of viscous flows against bodies with corners and edges by boundary element and multigrid methods. Numer. Math. 52, 81–99 (1987). https://doi.org/10.1007/BF01401023
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01401023