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On the Utilization of Edge Matrices in Algebraic Multigrid

  • J. K. Kraus
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3743)

Abstract

We are interested in the design of efficient algebraic multigrid (AMG) methods for the solution of large sparse systems of linear equations arising from finite element (FE) discretization of second-order elliptic partial differential equations (PDEs). In particular, we introduce the concept of so-called “edge matrices”, which–in the present context–are extracted from the individual element matrices. This allows for the construction of spectrally equivalent approximations of the original stiffness matrix that can be utilized in the framework of AMG.

The edge matrices give rise to modify the definition of “strong” and “weak” connections (edges), which provides a basis for selecting the coarse-grid nodes in algebraic multigrid methods. Moreover, a reproduction of edge matrices on coarse levels offers the opportunity to combine classical coarsening algorithms with effective (energy minimizing) interpolation principles involving small-sized “computational molecules” (small collections of edge matrices). This yields a flexible and robust new variant of AMG, which we refer to as AMGm.

Keywords

Element Matrice Algebraic Multigrid Method Spectral Condition Number Large Sparse System Edge Matrix 
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.

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References

  1. 1.
    Berman, A., Plemmons, R.J.: Nonnegative Matrices in Mathematical Sciences. Academic Press, New York (1979)zbMATHGoogle Scholar
  2. 2.
    Brandt, A.: Algebraic multigrid theory: the symmetric case. Appl. Math. Comput. 19, 23–56 (1986)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    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, 1570–1592 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Chartier, T., Falgout, R.D., Henson, V.E., Jones, J.E., Manteuffel, T., McCormick, S.F., Ruge, J.W., Vassilevski, P.S.: Spectral AMGe (ρAMGe). SIAM J. Sci. Comput. 25, 1–26 (2004)CrossRefMathSciNetGoogle Scholar
  5. 5.
    Cleary, A.J., Falgout, R.D., Henson, V.E., Jones, J.E., Manteuffel, T.A., Mc- Cormick, S.F., Miranda, G.N., Ruge, J.W.: Robustness and scalability of algebraic multigrid. SIAM J. Sci. Stat. Comput. 21, 1886–1908 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Haase, G., Langer, U., Reitzinger, S., Schöberl, J.: Algebraic multigrid methods based on element preconditioning. International Journal of Computer Mathematics 78, 575–598 (2004)CrossRefGoogle Scholar
  7. 7.
    Henson, V.E., Vassilevski, P.: Element-free AMGe: General algorithms for computing the interpolation weights in AMG. SIAM J. Sci. Comput. 23, 629–650 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Jones, J.E., Vassilevski, P.: AMGe based on element agglomeration. SIAM J. Sci. Comput. 23, 109–133 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Kraus, J.K., Schicho, J.: Algebraic multigrid based on computational molecules, 1: Scalar elliptic problems. RICAM-Report No. 2005–05, Linz. Submitted to Computing (2005)Google Scholar
  10. 10.
    Langer, U., Reitzinger, S., Schicho, J.: Symbolic methods for the element preconditioning technique. In: Winkler, F., Langer, U. (eds.) SNSC 2001. LNCS, vol. 2630, pp. 293–308. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  11. 11.
    Ruge, J.W., Stüben, K.: Efficient solution of finite difference and finite element equations by algebraic multigrid (AMG). In: Paddon, D.J., Holstein, H. (eds.) Multigrid Methods for Integral and Differential Equations. The Institute of Mathematics and Its Applications Conference, pp. 169–212. Clarendon Press, Oxford (1985)Google Scholar
  12. 12.
    Stüben, K.: Algebraic multigrid (AMG): experiences and comparisons. Appl. Math. Comput. 13, 419–452 (1983)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Vaněk, P., Brezina, M., Mandel, J.: Convergence of algebraic multigrid based on smoothed aggregation. Numer. Math. 88, 559–579 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Vaněk, P., Mandel, J., Brezina, M.: Algebraic multigrid based on smoothed aggregation for second and fourth order problems. Computing 56, 179–196 (1996)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • J. K. Kraus
    • 1
  1. 1.Johann Radon Institute for Computational and Applied MathematicsLinzAustria

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