Skip to main content

A robust and efficient multigrid method

Part II: Specific Contributions

Part of the Lecture Notes in Mathematics book series (LNM,volume 960)

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   54.99
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   69.95
Price excludes VAT (Canada)
  • 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

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  • G.P. Astrachancev, An iterative method of solving elliptic net problems. USSR Comp. Math. Math. Phys. 11, 2, 171–182, 1971.

    CrossRef  Google Scholar 

  • N.S. Bakhvalov, On the convergence of a relaxation method with natural constraints on the elliptic operator, USSR Comp. Math. Math. Phys. 6, no.5, 101–135, 1966.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • C. Börgers, Private communication: Mehrgitterverfahren für eine Mehrstellendiskretisierung der Poisson-Gleichung und für eine zweidimensionale singular gestörte Aufgabe. Diplomarbeit (prof. Trottenberg), 1981.

    Google Scholar 

  • D. Braess, The contraction number of a multigrid method for solving the Poisson equation. Num. Math. 37, 387–404, 1981.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • A. Brandt, Multi-level adaptive technique (MLAT) for fast numerical solution to boundary value problems. In: Proc. Third Int. Conf. Num. Meth. Fluid Dyn., Paris, 1972, Lect. Notes in Phys. 18, 82–89, Springer-Verlag, Berlin etc. 1973.

    MATH  Google Scholar 

  • A. Brandt, Multi-level adaptive solutions to boundary-value problems, Math. Comp. 31 333–390, 1977.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • A. Brandt, Multi-level adaptive solutions to singular perturbation problems. In: Numerical analysis of singular perturbation problems (P.W. Hemker, J.J. Miller, eds.), 53–142, Academic Press, New York, 1979.

    Google Scholar 

  • A. Brandt and N. Dinar: Multi-grid solution to elliptic flow problems. In: Numerical methods for partial differential equations (S.V. Parter, ed.), 53–149. New York etc., Academic Press, 1979.

    CrossRef  Google Scholar 

  • A.R. Curtis, On a property of some test equations for finite difference or finite element methods. IMA J. Num. Anal. 1, 369–375, 1981.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • R.P. Fedorenko, A relaxation method for solving elliptic difference equations, USSR Comp. Math. Math. Phys. 1, 1092–1096, 1962.

    CrossRef  MATH  Google Scholar 

  • H. Foerster, K. Stüben, U. Trottenberg, Non-standard multigrid techniques using checkered relaxation and intermediate grids. In: Elliptic problem solvers (M. Schulz, ed.), 285–300, Academic Press, New York etc., 1981.

    CrossRef  Google Scholar 

  • P.O. Frederickson, Fast approximate inversion of large sparse linear systems, Mathematics report 7-75, Lakehead University, Thunder Bay, Canada, 1975.

    Google Scholar 

  • I. Gustafsson, A class of first order factorization methods. BIT 18, 142–156, 1978.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • W. Hackbusch, A fast iterative method for solving Poisson’s equation in a general region. In: Numerical treatment of differential equations, Oberwolfach 1976 (R. Bulirsch, R.D. Grigorieff, J. Schröder, eds.). Lecture Notes in Math. 631, Springer-Verlag, Berlin etc., 1978a.

    Google Scholar 

  • W. Hackbusch, On the multigrid method applied to difference equations, Computing 20 pp. 291–306, 1978b.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • W. Hackbusch, On the convergence of multi-grid iterations. Beiträge zur Numerischen Mathematik 9, 213–239, 1981.

    MATH  Google Scholar 

  • P.W. Hemker, The incomplete LU-decomposition as a relaxation method in multi-grid algorithms. In: Boundary and interior layers — computational and asymptotic methods, Proceedings, Dublin 1980, (J.J.H. Miller, ed.), Boole-Press, Dublin 1980a.

    Google Scholar 

  • P.W. Hemker, Fourier analysis of gridfunctions, prolongations and restrictions. Mathematical Centre, 413 Kruislaan, Amsterdam, Report NW 93/80, 1980b.

    Google Scholar 

  • P.W. Hemker, Introduction to multigrid methods, Nieuw Archief v. Wiskunde 29, 71–101, 1981.

    MathSciNet  MATH  Google Scholar 

  • A.M. Il’in, Differencing scheme for a differential equation with a small parameter affecting the highest derivative. Math. Notes Acad. Sc. USSR 6, 596–602, 1969.

    CrossRef  MATH  Google Scholar 

  • R. Kettler, in the present volume (1982).

    Google Scholar 

  • R. Kettler and J.A. Meijerink, A multigrid method and a combined multigrid-conjugate gradient method for elliptic problems with strongly discontinuous coefficients in general domains. KSEPL, Volmerlaan 6, Rijswijk, The Netherlands, Publication 604, 1981.

    Google Scholar 

  • J.A. Meijerink and H.A. van der Vorst, An iterative solution method for linear systems of which the coefficient matrix is a symmetric M-matrix, Math. Comp. 31 pp. 148–162, 1977.

    MathSciNet  MATH  Google Scholar 

  • W.J.A. Mol, On the choice of suitable operators and parameters in multigrid methods, Report NW 107/81, Mathematical Centre, Amsterdam, 1981.

    MATH  Google Scholar 

  • R.A. Nicolaides, On some theoretical and practical aspects of multigrid methods. Math. Comp. 33, pp. 933–952, 1979.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • M. Ries, U. Trottenberg, G. Winter, A note on MGR methods. Universität Bonn, preprint no. 461, 1981.

    Google Scholar 

  • U. Schumann, Fast elliptic solvers and their application in fluid dynamics. In: W. Kollmann (ed.), Computational fluid dynamics, Hemisphere, Washington etc., 1980.

    Google Scholar 

  • K. Stüben, C.A. Thole, U. Trottenberg, private communication, 1982.

    Google Scholar 

  • H.A. van der Vorst, Iterative solution methods for certain sparse linear systems with a non-symmetric matrix arising from PDE-problems. J. Comp. Phys. 44, 1–20, 1981.

    CrossRef  MATH  Google Scholar 

  • E.L. Wachspress, A rational finite element basis, chapter 10. Academic Press, New York, 1975.

    MATH  Google Scholar 

  • P. Wesseling, Numerical solution of the stationary Navier-Stokes equations by means of a multiple grid method and Newton iteration. Report NA-18, Delft University of Technology, 1977.

    Google Scholar 

  • P. Wesseling, Theoretical and practical aspects of a multigrid method, Report NA-37, Delft University of Technology, 1980.

    Google Scholar 

  • P. Wesseling and P. Sonneveld, Numerical experiments with a multiple grid and a preconditioned Lanczos type method. In: Approximation methods for Navier-Stokes problems, Proceedings, Paderborn 1979 (R. Rautmann, ed.), Lecture Notes in Math. 771, 543–562, Springer-Verlag, Berlin etc. 1980.

    CrossRef  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 1982 Springer-Verlag

About this paper

Cite this paper

Wesseling, P. (1982). A robust and efficient multigrid method. In: Hackbusch, W., Trottenberg, U. (eds) Multigrid Methods. Lecture Notes in Mathematics, vol 960. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0069947

Download citation

  • DOI: https://doi.org/10.1007/BFb0069947

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-11955-5

  • Online ISBN: 978-3-540-39544-7

  • eBook Packages: Springer Book Archive