Computing and Visualization in Science

, Volume 7, Issue 3–4, pp 121–127

On an energy minimizing basis for algebraic multigrid methods

Regular article

Abstract

This paper is devoted to the study of an energy minimizing basis first introduced in Wan, Chan and Smith (2000) for algebraic multigrid methods. The basis will be first obtained in an explicit and compact form in terms of certain local and global operators. The basis functions are then proved to be locally harmonic functions on each coarse grid “element”. Using these new results, it is illustrated that this basis can be numerically obtained in an optimal fashion. In addition to the intended application for algebraic multigrid method, the energy minimizing basis may also be applied for numerical homogenization.

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References

  1. 1.
    Babuška, I., Osborn, J.E.: Generalized finite element methods: their performance and their relation to mixed methods. SIAM J. Numer. Anal. 20(3), 510–536 (1983)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Bramble, J.H., Pasciak, J.E., Wang, J.,Xu, J.: Convergence estimates for multigrid algorithms without regularity assumptions. Math. Comp. 57(195), 23–45 (1991)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Brandt, A.: Multiscale scientific computation. Six year research summary, 1999Google Scholar
  4. 4.
    Brandt, A., McCormick, S.F., Ruge, J.W.: Multigrid methods for differential eigenproblems. SIAM J. Sci. Statist. Comput. 4(2), 244–260 (1983)CrossRefGoogle Scholar
  5. 5.
    Brandt, A., McCormick, S.F., Ruge, J.W.: Algebraic multigrid (AMG) for sparse matrix equations. In: Evans, D.J. (ed.), Sparsity and Its Applications. Cambridge: Cambridge University Press 1984Google Scholar
  6. 6.
    Brezina, M., Cleary, A.J., Falgout, R.D., Henson, V.E., Jones, J.E. Manteuffel, T.A. McCormick, S.F., Ruge, J.W.: Algebraic multigrid based on element interpolation (AMGe). SIAM J. Sci. Comput. 22(5), 1570–1592 (electronic) (2000)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Chan, T.F., Xu, J., Zikatanov, L.: An agglomeration multigrid method for unstructured grids. In 10-th international conference on Domain Decomposition methods, volume 218 of Contemporary Mathematics, pp. 67–81. American Mathematical Society 1998Google Scholar
  8. 8.
    de Zeeuw, P.M.: Matrix–dependent prolongations and restrictions in a blackbox multigrid solver. J. Comput. Appl. Math. 33, 1–27 (1990)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Dendy, J.E.: Black box multigrid. J. Comput. Phys. 48, 366–386 (1982)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Efendiev, Y.R., Hou, T.Y., Wu, Z.-H.: Convergence of a nonconforming multiscale finite element method. SIAM J. Numer. Anal. 37(3), 888–910 (electronic) (2000)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Marini, D., Brezzi, F., Suli, E.: Residual free bubbles for the advection-diffusion problems: the general error analysis. Numer. Math. 85, 31–47 (2000)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Hughes, T.J.R., Brezzi, F., Franca, L.P., Russo, A.: b=∫g. Comput. Methods Appl. Mech. Engrg. 145, 329–339 (1997)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Franca, L.P., Russo, A.: Deriving upwinding, mass lumping and selective reduced integration by residual-free bubbles. Appl. Math. 9(5), 83–88 (1996)Google Scholar
  14. 14.
    Hou, T.Y., Wu, X.-H., Cai, Z.: Convergence of a multiscale finite element method for elliptic problems with rapidly oscillating coefficients. Math. Comp. 68(227), 913–943 (1999)CrossRefGoogle Scholar
  15. 15.
    Hughes, T.J.R.: Multiscale phenomena: Green’s functions, the Dirichlet-to-Neumann formulation, subgrid scale models, bubbles and the origins of stabilized methods. Comput. Methods Appl. Mech. Engrg. 127, 387–401 (1995)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Jones, J., Vassilevski, P.: AMGe based on element agglomeration. SIAM J. Sci. Comp. 23(1), 109–133 (2001)CrossRefGoogle Scholar
  17. 17.
    Xu, J., Zou, J.: Some nonoverlapping domain decomposition method. SIAM Review 40(4), 857–914 (1999)CrossRefGoogle Scholar
  18. 18.
    Mandel, J., Brezina, M., Vaněk, P.: Energy optimization of algebraic multigrid bases. Computing 62(3), 205–228 (1999)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Ruge, J.W., Stüben, K.: Algebraic multigrid. In: McCormick, S.F. (ed.), Multigrid methods, Frontiers in applied mathematics, pp. 73–130, Philadelphia, Pennsylvania: SIAM 1987Google Scholar
  20. 20.
    Vaněk, P., Mandel, J., Brezina, M.: Algebraic multigrid by smoothed aggregation for second and fourth order elliptic problems. Computing 56(3), 179–196 (1996), International GAMM-Workshop on Multi-level Methods (Meisdorf, 1994)CrossRefGoogle Scholar
  21. 21.
    Vaněk, P., Brezina, M., Mandel, J.: Convergence of algebraic multigrid based on smoothed aggregation. Numer. Math. 88(3), 559–579 (2001)Google Scholar
  22. 22.
    Wan, W.L.: Scalable and Multilevel Iterative Methods. PhD thesis, UCLA, Department of Mathematics, June 1998Google Scholar
  23. 23.
    Wan, W.L., Chan, T.F., Smith, B.: An energy-minimizing interpolation for robust multigrid methods. SIAM J. Sci. Comput. 21(4), 1632–1649 (electronic) (2000)CrossRefGoogle Scholar
  24. 24.
    Xu, J.: Iterative methods by space decomposition and subspace correction. SIAM Review 34, 581–613 (1992)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Xu, J., Zikatanov, L.: The method of subspace corrections and the method of alternating projections in Hilbert space. J. of Amer. Math. Soc. 15(3), 573–597 (2002)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  1. 1.Department of MathematicsThe Pennsylvania State UniversityUSA

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