Algebraic Multilevel Methods with Aggregations: An Overview

  • Radim Blaheta
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3743)


This paper deals with the numerical solution of elliptic boundary value problems by multilevel solvers with coarse levels created by aggregation. Strictly speaking, it deals with the construction of the coarse levels by aggregation, possible improvement of the simple aggregation technique and use of aggregations in multigrid, AMLI preconditioners and two-level Schwarz methods.


Multigrid Method Aggregation Technique Multilevel Method Schwarz Method Coarse Space 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Axelsson, O.: Iterative Solution Methods. Cambridge Univ. Press, Cambridge (1994)zbMATHGoogle Scholar
  2. 2.
    Blaheta, R.: A multilevel method with correction by aggregation for solving discrete elliptic problems. Aplikace Matematiky 31, 365–378 (1986)zbMATHMathSciNetGoogle Scholar
  3. 3.
    Blaheta, R.: Iterative methods for solving problems of elasticity, Thesis, Charles University, Prague (1987) (in Czech) Google Scholar
  4. 4.
    Blaheta, R.: A multilevel method with overcorrection by aggregation for solving discrete elliptic problems. J. Comp. Appl. Math. 24, 227–239 (1988)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Blaheta, R.: GPCG generalized preconditioned CG method and its use with non-linear and non-symmetric displacement decomposition preconditioners. Numer. Linear Algebra Appl. 9, 527–550 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Blaheta, R.: Space decomposition preconditioners and parallel solver. In: Feistauer, M., et al. (eds.) Numerical Mathematics and Advanced Applications (ENUMATH 2003), pp. 20–38. Springer, Berlin (2004)Google Scholar
  7. 7.
    Blaheta, R.: An AMLI preconditioner with hierarchical decomposition by aggregation (in progress)Google Scholar
  8. 8.
    Blaheta, R., Neytcheva, M., Margenov, S.: Uniform estimate of the constant in the strengthened CBS inequality for anisotropic non-conforming FEM systems. Numerical Linear Algebra with Applications 11, 309–326 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Braess, D.: Towards Algebraic Multigrid for Elliptic Problems of Second Order. Computing 55, 379–393 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Brezina, M.: Robust iterative methods on unstructured meshes, PhD. thesis, University of Colorado at Denver (1997)Google Scholar
  11. 11.
    Brezina, M., Falgout, R., MacLachlan, S., Manteuffel, T., McCormick, S., Ruge, J.: Adaptive Smoothed Aggregation (áSA). SIAM J. Sci. Comput. 25, 1896–1920 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Brezina, M., Tong, C., Becker, R.: Parallel Algebraic Multigrids for Structural Mechanics. SIAM J. Sci. Comput. (2004) (to appear); Also available as LLNL Technical Report UCRL-JRNL-204167Google Scholar
  13. 13.
    Briggs, W., Henson, V., McCormick, S.: A Multigrid tutorial, 2nd edn. SIAM, Philadelphia (2000)zbMATHCrossRefGoogle Scholar
  14. 14.
    Bulgakov, V.E., Kuhn, G.: High-Performance Multilevel Iterative Aggregation Solver for Large Finite Element Structural Analysis Problems. Int. J. Numer. Methods in Engrg. 38, 3529–3544 (1995)zbMATHCrossRefGoogle Scholar
  15. 15.
    Fish, J., Belsky, V.: Generalized Aggregation Multilevel Solver. Int. J. for Numerical Methods in Engineering 41, 4341–4361 (1997)CrossRefMathSciNetGoogle Scholar
  16. 16.
    Griebel, M., Oeltz, D., Schweitzer, M.A.: An Algebraic Multigrid Method for Linear Elasticity. SIAM J. Sci. Computing 25, 385–407 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Jenkins, E.W., Kelley, C.T., Miller, C.T., Kees, C.E.: An aggregation-based domain decomposition preconditioner for groundwater flow. SIAM J. Sci. Comput. 23, 430–441 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Notay, Y.: Robust parameter free algebraic multilevel preconditioning. Numer. Lin. Alg. Appl. 9, 409–428 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Notay, Y.: Aggregation-based algebraic multilevel preconditioning. SIAM J. Matrix Anal. Appl. (2005) (to appear)Google Scholar
  20. 20.
    Notay, Y.: Algebraic multigrid and algebraic multilevel methods: a theoretical comparison. Numer. Lin. Alg. Appl. 12, 419–451 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Sala, M., Shadid, J., Tuminaro, R.: An Improved Convergence Bound for Aggregation- Based Domain Decomposition Preconditioners. Submitted to SIMAX (2005)Google Scholar
  22. 22.
    Southwell, R.S.: Relaxation Methods in Theoretical Physics. Oxford University Press, Oxford (1946)Google Scholar
  23. 23.
    Stüben, K.: Algebraic Multigrid (AMG): An Introduction with Applications. GMD Report 53 (March 1999); Multigrid, T., et al.: An Introduction to Algebraic Multigrid, Appendix A, pp. 413–532. Academic Press, London (2001)Google Scholar
  24. 24.
    Toselli, A., Widlund, O.: Domain Decomposition Methods— Algorithms and Theory. Springer Series in Computational Mathematics, vol. 34. Springer, Heidelberg (2005)zbMATHGoogle Scholar
  25. 25.
    Vanek, P.: Acceleration of Convergence of a Two Level Algorithm by Smooth Transfer Operators. Appl. Math. 37, 265–274 (1992)zbMATHMathSciNetGoogle Scholar
  26. 26.
    Vanek, P., Mandel, J., Brezina, M.: Algebraic multigrid by smoothed aggregation for second and fourth order elliptic problems. Computing 56, 179–196 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Vanek, P., Brezina, M., Mandel, J.: Convergence of algebraic multigrid based on smoothed aggregation. Numerische Mathematik 88, 559–579 (2001)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Radim Blaheta
    • 1
  1. 1.Department of Applied MathematicsInstitute of Geonics AS CROstrava-PorubaCzech Republic

Personalised recommendations