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An algebraic multigrid method for finite element systems on criss-cross grids

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Abstract

In this paper, we design and analyze an algebraic multigrid method for a condensed finite element system on criss-cross grids and then provide a convergence analysis. Criss-cross grid finite element systems represent a large class of finite element systems that can be reduced to a smaller system by first eliminating certain degrees of freedoms. The algebraic multigrid method that we construct is analogous to many other algebraic multigrid methods for more complicated problems such as unstructured grids, but, because of the specialty of our problem, we are able to provide a rigorous convergence analysis to our algebraic multigrid method.

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References

  1. D.N. Arnold and J. Qin, Quadratic velocity/linear pressure stokes element, in: Advances in Computer Methods for Partial Differential Equations, Vol. VII, eds. G. Richter, R. Vichnevetsky and D. Knight (IMACS, 1992).

  2. I.M. Babuska, FEM-Latest Developments, Open Problems and Perspectives, Internat. Colloquium on Applications of Computer Science and Mathematics in Architecture and Civil Engineering, Weimar, Germany (26 February–1 March, 1997).

  3. R. Bank and J. Xu, An algorithm for coarsening unstructured meshes, Numer. Math. 73(1) (1996) 1–36.

    Article  MathSciNet  Google Scholar 

  4. J.H. Bramble, J.E. Pasciak, J. Wang and J. Xu, Convergence estimates for multigrid algorithms without regularity assumptions, Math. Comp. 57(195) (1991) 23–45.

    Article  MathSciNet  Google Scholar 

  5. A. Brandt, Multiscale scientific computation, Six year research summary (1999).

  6. A. Brandt, S.F. McCormick and J.W. Ruge, Algebraic multigrid (AMG) for sparse matrix equations, in: Sparsity and Its Applications, ed. D.J. Evans (Cambridge Univ. Press, Cambridge, 1984).

    Google Scholar 

  7. M. Brezina, A.J. Cleary, R.D. Falgout, V.E. Henson, J.E. Jones, T.A. Manteuffel, S.F. McCormick and J.W. Ruge, Algebraic multigrid based on element interpolation (AMGe), SIAM J. Sci. Comput. 22(5) (2000) 1570–1592 (electronic).

    Article  MathSciNet  Google Scholar 

  8. F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods (Spinger, New York, 1991).

    Google Scholar 

  9. T.F. Chan, J. Xu and L. Zikatanov, An agglomeration multigrid method for unstructured grids, in: 10th Internat. Conf. on Domain Decomposition Methods, Contemporary Mathematics, Vol. 218 (Birkhäuser, Boston, 1998) pp. 67–81.

    Google Scholar 

  10. P.M. de Zeeuw, Matrix-dependent prolongations and restrictions in a black box multigrid solver, J. Comput. Appl. Math. 33 (1990) 1–27.

    Article  MATH  MathSciNet  Google Scholar 

  11. J.E. Dendy, Black box multigrid, J. Comput. Phys. 48 (1982) 366–386.

    Article  MATH  MathSciNet  Google Scholar 

  12. Y.R. Efendiev, T.Y. Hou and Z.-H. Wu, Convergence of a nonconforming multiscale finite element method, SIAM J. Numer. Anal. 37(3) (2000) 888–910 (electronic).

    Article  MathSciNet  Google Scholar 

  13. J. Mandel, M. Brezina and P. Vanĕk, Energy optimization of algebraic multigrid bases, Computing 62(3) (1999) 205–228.

    Article  MathSciNet  Google Scholar 

  14. J. Ruge and K. Stüben, Algebraic multigrid, in: Multigrid Methods, ed. S. McCormick (SIAM, Philadelphia, PA, 1987).

    Google Scholar 

  15. K. Stüben, A review of algebraic multigrid, GMD Report 69 (1999).

  16. P. Vanek, J. Mandel and M. Brezina, Algebraic multigrid by smoothed aggregation for second and fourth order elliptic problems, Computing 56(3) (1996) 179–196.

    Article  MathSciNet  Google Scholar 

  17. P. Vanek, M. Brezina and J. Mandel, Convergence of algebraic multigrid based on smoothed aggregation, Numer. Math. 88(3) (2001) 559–579.

    Article  MathSciNet  Google Scholar 

  18. W.L. Wan, T.F. Chan and B. Smith, An energy-minimizing interpolation for robust multigrid methods, SIAM J. Sci. Comput. 21(3) (2000) 559–579.

    MathSciNet  Google Scholar 

  19. J. Xu, Iterative methods by space decomposition and subspace correction, SIAM Rev. 34 (1992) 581–613.

    Article  MATH  MathSciNet  Google Scholar 

  20. J. Xu and L. Zikatanov, The method of alternating projections and the method of subspace corrections on Hilbert space, J. Amer. Math. Soc. 15 (2002) 1429–1446.

    Article  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. Institute for Computational and Applied Mathematics of Xiangtan University, China

    Shi Shu & Haiyuan Yu

  2. Department of Mathematics and Center for Computational Mathematics and Application of Pennsylvania State University, USA

    Jinchao Xu

  3. Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and System Sciences, Chinese Academy of Sciences, China

    Ying Yang

Authors
  1. Shi Shu
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  2. Jinchao Xu
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  3. Ying Yang
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  4. Haiyuan Yu
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Additional information

Communicated by A. Zhou

Dedicated to Professor Charles A. Micchelli on the occasion of his 60th birthday

The work was supported in part by NSAF(10376031) and National Major Key Project for basic researches and by National High-Tech ICF Committee in China.

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Shu, S., Xu, J., Yang, Y. et al. An algebraic multigrid method for finite element systems on criss-cross grids. Adv Comput Math 25, 287–304 (2006). https://doi.org/10.1007/s10444-004-7627-y

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  • DOI: https://doi.org/10.1007/s10444-004-7627-y

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