Hostname: page-component-586b7cd67f-2brh9 Total loading time: 0 Render date: 2024-11-27T14:11:50.377Z Has data issue: false hasContentIssue false
  • English
  • Français

Old and new convergence proofs for multigrid methods

Published online by Cambridge University Press:  07 November 2008

Harry Yserentant
Harry Yserentant
Affiliation:
Mathematisches InstitutUniversität Tübingen, D-7400 Tübingen, Germany, E-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Multigrid methods are the fastest known methods for the solution of the large systems of equations arising from the discretization of partial differential equations. For self-adjoint and coercive linear elliptic boundary value problems (with Laplace's equation and the equations of linear elasticity as two typical examples), the convergence theory reached a mature, if not its final state. The present article reviews old and new developments for this type of equation and describes the recent advances.

Type
Research Article
Information
Acta Numerica , Volume 2 , January 1993 , pp. 285 - 326
Copyright
Copyright © Cambridge University Press 1993

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Axelsson, O. and Gustafsson, I. (1983), ‘Preconditioning and two-level multigrid methods of arbitrary degree of approximation’, Math. Comput. 40, 219–242.CrossRefGoogle Scholar
Axelsson, O. and Vassilevski, P. (1989), ‘Algebraic multilevel preconditioning methods I’, Numer. Math. 56, 157–177.CrossRefGoogle Scholar
Bachvalov, N.S. (1966), ‘On the convergence of a relaxation method with natural constraints on the elliptic operator’ (in Russian), USSR Comput. Math, and Math. Phys. 6.5, 101–135.CrossRefGoogle Scholar
Bänsch, E. (1991), ‘Local mesh refinement in 2 and 3 dimensions’, Impact of Computing in Sci. Engrg. 2, 181–191.CrossRefGoogle Scholar
Bank, R.E. (1981), ‘A comparison of two multilevel iterative methods for nonsymmetric and indefinite elliptic finite element equations’, SIAM J. Numer. Anal. 18, 724–743.CrossRefGoogle Scholar
Bank, R.E. (1990), PLTMG: A Software Package for Solving Elliptic Partial Differential Equations, SIAM (Philadelphia).Google Scholar
Bank, R.E. and Douglas, C.C. (1985), ‘Sharp estimates for multigrid convergence rates of convergence with general smoothing and acceleration’, SIAM J. Numer. Anal. 22, 617–633.CrossRefGoogle Scholar
Bank, R.E. and Dupont, T. (1980), ‘Analysis of a two-level scheme for solving finite element equations’, Report CNA-159, Center for Numerical Analysis, University of Texas at Austin.Google Scholar
Bank, R.E. and Dupont, T. (1981), ‘An optimal order process for solving finite element equations’, Math. Comput. 36, 35–51.CrossRefGoogle Scholar
Bank, R.E. and Rose, D.J. (1982), ‘Analysis of a multilevel iterative method for nonlinear finite element equations’, Math. Comput. 39, 453–465.CrossRefGoogle Scholar
Bank, R.E. and Weiser, A. (1985), ‘Some a-posteriori estimators for elliptic partial differential equations’, Math. Comput. 44, 283–301.CrossRefGoogle Scholar
Bank, R.E., Dupont, T. and Yserentant, H. (1988), ‘The hierarchical basis multigrid method’, Numer. Math. 52, 427–458.CrossRefGoogle Scholar
Bank, R.E., Welfert, B. and Yserentant, H. (1990), ‘A class of iterative methods for solving saddle point problems’, Numer. Math. 56, 645–666.CrossRefGoogle Scholar
Björstad, P.E. and Mandel, J. (1991), ‘On the spectra of sums of orthogonal projections with applications to parallel computing’, BIT 31, 76–88.CrossRefGoogle Scholar
Bornemann, F.A. and Yserentant, H. (1992), ‘A basic norm equivalence for the theory of multilevel methods’, Numer. Math., to appear.Google Scholar
Braess, D. (1981), ‘The contraction number of a multigrid method for solving the Poisson equation’, Numer. Math. 37, 387–404.CrossRefGoogle Scholar
Braess, D. and Hackbusch, W. (1983), ‘A new convergence proof for the multigrid method including the V-cycle’, SIAM J. Numer. Anal. 20, 967–975.CrossRefGoogle Scholar
Bramble, J.H. and Pasciak, J.E. (1988), ‘A preconditioning technique for indefinite systems resulting from mixed approximations of elliptic problems’, Math. Comput. 181, 1–17.CrossRefGoogle Scholar
Bramble, J.H. and Pasciak, J.E. (1991), ‘New estimates for multilevel algorithms including the V-cycle’, Technical Report '91–47, Mathematical Sciences Institute, Cornell University.Google Scholar
Bramble, J.H., Pasciak, J.E., Wang, J. and Xu, J. (1991a), ‘Convergence estimates for product iterative methods with applications to domain decomposition’, Math. Comput. 57, 1–21.CrossRefGoogle Scholar
Bramble, J.H., Pasciak, J.E., Wang, J. and Xu, J. (1991b), ‘Convergence estimates for multigrid algorithms without regularity assumptions’, Math. Comput. 57, 23–45.CrossRefGoogle Scholar
Bramble, J.H., Pasciak, J.E. and Xu, J. (1990), ‘Parallel multilevel preconditioners’, Math. Comput. 55, 1–22.CrossRefGoogle Scholar
Brandt, A. (1977), ‘Multi-level adaptive solutions to boundary-value problems’, Math. Comput. 31, 333–390.CrossRefGoogle Scholar
Brandt, A. (1982), ‘Guide to multigrid development’, in Multigrid Methods (Hackbusch, W. and Trottenberg, U., eds), Lecture Notes in Mathematics 960, Springer (Berlin, Heidelberg, New York).Google Scholar
Chan, T., Glowinski, R., Periaux, J. and Widlund, O.B., eds (1989), Domain Decomposition Methods, Proceedings, Los Angeles 1988, SIAM (Philadelphia).Google Scholar
Ciarlet, P.G. (1978), The Finite Element Method for Elliptic Problems, North-Holland (Amsterdam).Google Scholar
Dahmen, W. and Kunoth, A. (1992), ‘Multilevel preconditioning’, Numer. Math., to appear.CrossRefGoogle Scholar
Deuflhard, P. (1992), Newton-Techniques for Highly Nonlinear Problems- Theory and Algorithms, Academic Press (New York) in preparation.Google Scholar
Deuflhard, P., Leinen, P. and Yserentant, H. (1989), ‘Concepts of an adaptive hierarchical finite element code’, Impact of Computing in Sci. Engrg 1, 3–35.CrossRefGoogle Scholar
Dryja, M. and Widlund, O. (1991), ‘Multilevel additive methods for elliptic finite element problems’, in Parallel Algorithms for Partial Differential Equations, Proceedings, Kiel 1990 (Hackbusch, W., ed.), Vieweg, Braunschweig (Wiesbaden).Google Scholar
Fedorenko, R.P. (1964), ‘The speed of convergence of an iterative process’ (in Russian), USSR Comput. Math, and Math. Phys. 4.2, 227–235.CrossRefGoogle Scholar
Freund, R.W., Golub, G.H. and Nachtigal, N.M. (1992), ‘Iterative solution of linear systems’, Acta Numerica 1992, 57–100.Google Scholar
Hackbusch, W. (1976), ‘Ein iteratives Verfahren zur schnellen Auflösung elliptischer Randwertprobleme’, Report 76–12, Mathematisches Institut der Universität zu Köln.Google Scholar
Hackbusch, W. (1981), ‘On the convergence of multi-grid iterations’, Beiträge Numer. Math. 9, 213–329.Google Scholar
Hackbusch, W. (1982), ‘Multi-grid convergence theory’, in Multigrid Methods (Hackbusch, W. and Trottenberg, U., eds), Lecture Notes in Mathematics 960, Springer (Berlin, Heidelberg, New York).CrossRefGoogle Scholar
Hackbusch, W. (1985), Multigrid Methods and Applications, Springer (Berlin, Heidelberg, New York).CrossRefGoogle Scholar
Hackbusch, W. (1989a), ‘The frequency decomposition multi-grid algorithm’, in Robust Multi-Grid Methods, Proceedings Kiel 1988 (Hackbusch, W., ed.). Vieweg, Braunschweig (Wiesbaden).Google Scholar
Hackbusch, W. (1989b), ‘The frequency decomposition multi-grid method. Part I: Application to anisotropic equations’, Numer. Math. 56, 229–245.CrossRefGoogle Scholar
Hackbusch, W. (1991), Iterative Lösung großer schwachbesetzter Systeme, Teubner (Stuttgart) (English translation in preparation).CrossRefGoogle Scholar
Hackbusch, W. and Reusken, A. (1989), ‘Analysis of a damped nonlinear multilevel method’, Numer. Math. 55, 225–246.CrossRefGoogle Scholar
Hackbusch, W. and Trottenberg, U., eds (1982), Multigrid Methods, Proceedings, Köln 1981, Lecture Notes in Mathematics 960, Springer (Berlin, Heidelberg, New York).Google Scholar
Mandel, J., McCormick, S. and Bank, R. (1987), ‘Variational multigrid theory’, in Multigrid Methods (McCormick, S., ed.), SIAM (Philadelphia).Google Scholar
McCormick, S., ed. (1987), Multigrid Methods, SIAM (Philadelphia).CrossRefGoogle Scholar
Oswald, P. (1990), ‘On function spaces related to finite element approximation theory’, Zeitschrift für Analysis und ihre Anwendungen 9, 43–64.CrossRefGoogle Scholar
Oswald, P. (1991), ‘On discrete norm estimates related to multilevel preconditioners in the finite element method’, in Proceedings of the International Conference on the Constructive Theory of Functions, Varna 1991.Google Scholar
Oswald, P. (1992), ‘Stable splittings of Sobolev spaces and fast solution of variational problems’, Forschungsergebnisse der Friedrich-Schiller-Universität Jena, Math/92/5.Google Scholar
Reusken, A. (1992), ‘On maximum norm convergence of multigrid methods for elliptic boundary value problems’, Report RANA 92–03, Department of Mathematics and Computer Science, Eindhoven University.Google Scholar
Rivara, M.C. (1984), ‘Algorithms for refining triangular grids suitable for adaptive and multigrid techniques’, Int. J. Numer. Meth. Engrg 20, 745–756.CrossRefGoogle Scholar
Schatz, A. (1974), ‘An observation concerning Ritz–Galerkin methods with indefinite bilinear forms’, Math. Comput. 28, 959–962.CrossRefGoogle Scholar
Stevenson, R. (1992), ‘New estimates of the contraction number of V-cycle multigrid with applications to anisotropic equations’, Preprint Nr. 713, Department of Mathematics, University of Utrecht.Google Scholar
Stüuben, K. and Trottenberg, U. (1982), ‘Multigrid methods: fundamental algorithms, model problem analysis and applications’, in Multigrid Methods (Hackbusch, W. and Trottenberg, U., eds), Lecture Notes in Mathematics 960, Springer (Berlin, Heidelberg, New York).Google Scholar
Varga, R.S. (1962), Matrix Iterative Analysis, Prentice Hall (Englewood Cliffs, NJ).Google Scholar
Vassilevski, P. (1992), ‘Preconditioning nonsymmetric and indefinite finite element matrices’, J. Numer. Linear Alg. with Appl. 1, 59–76.Google Scholar
Wesseling, P. (1992), An Introduction to Multigrid Methods, John Wiley (Chichester).Google Scholar
Widlund, O. (1989), ‘Optimal iterative refinement methods’, in Domain Decomposition Methods (Chan, T. et al. , eds), SIAM (Philadelphia).Google Scholar
Wittum, G. (1989a), ‘Linear iterations as smoothers in multigrid methods: theory with applications to incomplete decompositions’, Impact of Computing in Sci. Engrg 1, 180–215.CrossRefGoogle Scholar
Wittum, G. (1989b), ‘On the robustness of ILU smoothing’, SIAM J. Sci. Statist. Comput. 10, 699–717.CrossRefGoogle Scholar
Xu, J. (1989), ‘Theory of multilevel methods’, Report AM48, Department of Mathematics, Pennsylvania State University.Google Scholar
Xu, J. (1992a), ‘A new class of iterative methods for nonsymmetric and indefinite problems’, SIAM J. Numer. Anal. 29, 303–319.CrossRefGoogle Scholar
Xu, J. (1992b),‘Iterative methods by space decomposition and subspace correction’, SIAM Rev. 34.CrossRefGoogle Scholar
Young, D.M. (1950), ‘Iterative methods for solving partial differential equations of elliptic type’, Thesis, Harvard University.Google Scholar
Yserentant, H. (1983), ‘On the convergence of multi-level methods for strongly nonuniform families of grids and any number of smoothing steps per level’, Computing 30, 305–313.CrossRefGoogle Scholar
Yserentant, H. (1986a), ‘The convergence of multi-level methods for solving finite element equations in the presence of singularities’, Math. Comput. 47, 399–409.Google Scholar
Yserentant, H. (1986b), ‘On the multi-level splitting of finite element spaces’, Numer. Math. 49, 379–412.CrossRefGoogle Scholar
Yserentant, H. (1986c), ‘On the multi-level splitting of finite element spaces for indefinite elliptic boundary value problems’, SIAM J. Numer. Anal. 23, 581–595.CrossRefGoogle Scholar
Yserentant, H. (1988), ‘Preconditioning indefinite discretization matrices’, Numer. Math. 54, 719–734.CrossRefGoogle Scholar
Yserentant, H. (1990), ‘Two preconditioners based on the multi-level splitting of finite element spaces’, Numer. Math. 58, 163–184.CrossRefGoogle Scholar
Yserentant, H. (1992), ‘Hierarchical bases’, in ICIAM 91 (O'Malley, R.E., ed.), SIAM (Philadelphia).Google Scholar
Zhang, S. (1990), ‘Optimal order non-nested multigrid methods for solving finite element equations II: on non-quasiuniform meshes’, Math. Comput. 55, 439–450.Google Scholar
Zhang, X. (1991), ‘Multilevel additive Schwarz methods’, Technical Report 582, Courant Institute, New York.Google Scholar
Zhang, X. (1992), ‘Multilevel Schwarz methods’, Numer. Math., to appear.CrossRefGoogle Scholar