Skip to main content
SpringerLink
Log in

Theoretical and Practical Aspects of Multigrid Methods in Boundary Element Calculations

  • Chapter
Computational Aspects

Part of the book series: Topics in Boundary Element Research ((TBOU,volume 3))

  • 93 Accesses

Summary

In this paper multigrid methods are advocated for the fast solution of the large nonsparse systems of equations that occur in boundary-element methods. Multigrid methods combine relaxation schemes and coarse-grid corrections. Ample attention is given to the decomposition of the system matrix in order to obtain a relaxation scheme that reduces the high-frequency components of the iteration error. It is shown that the decomposition should take the edges of the boundary into account, because they have a strong influence on the smoothing property of the relaxation scheme. The practical aspects of the multigrid method are concerned with the use of the method in boundary element calculations. The choice of the coarse-grid operators, the interactions between the grids and the implementation of the algorithm are discussed. The theoretical investigations show that the multigrid method converges more rapidly as the number of boundary elements increases. This is illustrated for two plane problems: (1) potential flow around an aerofoil and (2) interior fundamental problem of elasticity.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Similar content being viewed by others

References

  1. Alund, A., Iterative methods to compute the singularity distribution in three-dimensional panel methods, Report L-0–1 R 97, SAAB-Scania, Linköping, Sweden, 1984

    Google Scholar 

  2. Anselone, P.M., Collectively compact operator approximation theory, Englewood Cliffs, New Jersey, Prentice-Hall, 1971

    Google Scholar 

  3. Barnard, A.C.L., Duck, I.M., Lynn, M.S., and Timlake, W.P., The application of electromagnetic theory to electrocardiology, II. Numerical solution of the integral equations. Biophys. J. 7, 463–491, 1967

    Google Scholar 

  4. Berkhoff, J.C.W., Diffraction of water waves. In: Colloquium Numerical Treatment of Integral Equations (H.J.J. to Ride, ed), MC-Syllabus 41, Mathematisch Centrum, Amsterdam 1979, pp. 241–258

    Google Scholar 

  5. Brandt, A., Multi-level adaptive solutions to boundary-valve problems. Mathematics of Computations 31, 333–390, 1977

    Article  MATH  MathSciNet  Google Scholar 

  6. Brebbia, C. A., The boundary element method for engineers. Pentech Press, London 1978

    Google Scholar 

  7. Brebbia, C.A. and Walker, S., Boundary element techniques in Engineering. Butterworth, London 1980

    MATH  Google Scholar 

  8. Costabel, M. and 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

    Google Scholar 

  9. Edwards, T.W. and Van Bladel, K., Electrostatic dipole moment of a dielectric cube. Applied Scientific Research 9, 151–155, 1961

    Article  Google Scholar 

  10. Filippi, P., Layer potentials and acoustic diffraction. J. Sound and Vibration 54, 473–500, 1977

    Article  MATH  ADS  Google Scholar 

  11. Hackbusch, W., Die schnelle Auflösung der Fredholmschen Integralgleichung zweiter Art. Beiträge zur Numerischen Mathematik 9, 47–62, 1981

    Google Scholar 

  12. Hackbusch, W., Multigrid convergence theory, Multigrid Methods, Proceedings, Köln 1981 (W. Hackbusch and Trottenberg, U., eds.). Lecture Notes in Mathematics, 960, pp. 177–219, Springer-Verlag, Berlin, Heidelberg, New York 1982

    Google Scholar 

  13. Hebeker, F.K, On a multigrid method to solve the integral equations of 3-D Stokes’ flow, Efficient Solutions of Elliptic Systems. Proceedings, Kiel 1984 (W. Hackbusch, ed.). Vieweg Verlag, Braunschweig/Wiesbaden 1984

    Google Scholar 

  14. Hemker, P.W. and Schippers, H., Multiple grid methods for the solution of Fredhom integral equations of the second kind. Math. Comp. 36, 215–232, 1981

    Article  MATH  MathSciNet  Google Scholar 

  15. Hess, J.L. and Smith. A.M.O., Calculation of potential flow about arbitrary bodies. Progress in Aero Science 8, 1–138, 1967

    Article  MATH  ADS  Google Scholar 

  16. Hsiao, G.C., Kopp, P., and Wendland, W.L., A Galerkin collocation method for some integral equations of the first kind. Computing 25, 89–130, 1980

    Article  MATH  MathSciNet  Google Scholar 

  17. Hsiao, G.C., Kopp, P., and Wendland, W.L., Some applications of a Galerkin-collocation method for boundary integral equations of the first kind. Preprint Nr. 768, Technical University Darmstadt, Dept. Mathematics, 1983

    Google Scholar 

  18. Hsiao, G.C. and Maccamy, R.C., Solution of boundary value problems by integral equations of the first kind. SIAM Review 15, 687–705, 1973

    Article  MATH  MathSciNet  Google Scholar 

  19. Hsiao, G.C. and Wendland, W.L., A finite element method for some integral equations of the first kind. J. Math. Anal. Appl. 58, 449–481, 1977

    Article  MATH  MathSciNet  Google Scholar 

  20. Jaswon, M. A. and Symm, G.T., Integral equation methods in potential theory and elastostatics. Academic Press, London 1977

    MATH  Google Scholar 

  21. Jones, D.S., Integral equations for the exterior acoustic problem. Quart. J. Mech. Appl. Math. 27, 129–142, 1974

    Article  MATH  MathSciNet  Google Scholar 

  22. Martensen, E., Berechnung der Druckverteilung an Gitterprofilen in ebener Potentialströ mung mit einer Fredholmschen Integralgleichung. Arch, Rat. Mech. Anal. 3, 235–270, 1959

    Article  MATH  MathSciNet  Google Scholar 

  23. Muschelischwili, N.I., Singuläre Integralgleichungen. Akademie-Verlag, Berlin 1965

    MATH  Google Scholar 

  24. NAG Fortran Library Manual mark 10, Numerical Algorithms Group. Oxford 1983

    Google Scholar 

  25. Nedelec, J.C., Curved finite element methods for the solution of singular integral equations on surfaces in R3. Comp. Math. Appl. Mech. Eng. 8, 61–80, 1976

    Article  MATH  MathSciNet  Google Scholar 

  26. Nowak, Z.P., Use of the multigrid method for Laplacian problems in three dimensions, Multigrid Methods, Proceedings, Köln 1981 (W. Hackbusch and U. Trottenberg, eds.). Lecture Notes in Mathematics, 960, pp. 576–598, Springer-Verlag, Berlin, Heidelberg, New York 1982

    Google Scholar 

  27. Oskam, B. and Fray, J.M.J., General relaxation schemes in multigrid algorithms for higher order singularity methods. J. Comp. Physics 48, 423–440, 1982

    Article  MATH  ADS  MathSciNet  Google Scholar 

  28. Rizzo, -F.J., An integral equation to boundary value problems of classical elastostatics. Q. Appl. Math. 25, 83–95, 1967

    MATH  Google Scholar 

  29. Schippers, H., On the regularity of the principal value of the double-layer potential. J. Engineering Math. 16, 59–76, 1982

    Article  MATH  MathSciNet  Google Scholar 

  30. Schippers, H., Application of multigrid methods for integral equations to two problems from fluid dynamics. J. Comp. Physics 48, 441–461, 1982

    Article  MATH  ADS  Google Scholar 

  31. 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

    Google Scholar 

  32. Schippers, H., Multigrid methods for boundary integral equations, NLR MP 82059 U, National Aerospace Laboratory, Amsterdam 1982, Numerische Mathematik

    Google Scholar 

  33. Trottenberg, U., Multigrid methods: fundamental algorithms, model problem analysis and applications, Multigrid Methods, Proceedings, Köln 1981 (W. Hackbusch and U. Trottenberg, eds.). Lecture Notes in Mathematics, 960, pp. 1–176, Springer-Verlag, Berlin, Heidelberg, New York 1982

    Google Scholar 

  34. Wendland, W.L., On Galerkin collocation methods for integral equations of elliptic boundary value problems. In: Numerical Treatment of Integral Equations (J. Albrecht and L. Collatz, eds.). Intern. Ser. Numer. Math., 53, pp. 244–275, Birkhäuser, Basel 1980

    Google Scholar 

  35. 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

    Google Scholar 

  36. Wendland, W.L., Boundary element methods and their asymptotic convergence, Lecture Notes of the CISM Summer-School on Theoretical Acoustics and Numerical Techniques, International Centre for Mechanical Sciences, Udine (Italy), 1982 (P. Filippi, ed.), to appear in: Lecture Notes in Physics, Springer-Verlag, Berlin, Heidelberg, New York 1983

    Google Scholar 

  37. Wesseling, P., A robust and efficient multigrid method, Multigrid Methods, Proceedings, Köln 1981 (W. Hackbusch and U. Trottenberg, eds.), Lecture Notes in Mathematics, 960, pp. 614–630, Springer-Verlag, Berlin, Heidelberg, New York 1982

    Google Scholar 

  38. Wolff, H., Multiple grid method for the calculation of potential flow around 3-D bodies, Preprint NW 119/82, Dept. of Numerical Mathematics, Mathematical Centre, Amsterdam 1982

    Google Scholar 

Download references

Authors
  1. H. Schippers
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Computational Mechanics Institute, Wessex Institute of Technology, 52 Henstead Road, SO1 2DD, Southampton, England

    Carlos A. Brebbia

Rights and permissions

Reprints and permissions

Copyright information

© 1987 Springer-Verlag Berlin, Heidelberg

About this chapter

Cite this chapter

Schippers, H. (1987). Theoretical and Practical Aspects of Multigrid Methods in Boundary Element Calculations. In: Brebbia, C.A. (eds) Computational Aspects. Topics in Boundary Element Research, vol 3. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-82663-4_6

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-82663-4_6

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-82665-8

  • Online ISBN: 978-3-642-82663-4

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation