An algebraic multigrid method for finite element systems on crisscross grids
 Shi Shu,
 Jinchao Xu,
 Ying Yang,
 Haiyuan Yu
 … show all 4 hide
Abstract
In this paper, we design and analyze an algebraic multigrid method for a condensed finite element system on crisscross grids and then provide a convergence analysis. Crisscross 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.
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 HighTech ICF Committee in China.
 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).
 I.M. Babuska, FEMLatest 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).
 R. Bank and J. Xu, An algorithm for coarsening unstructured meshes, Numer. Math. 73(1) (1996) 1–36. CrossRef
 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. CrossRef
 A. Brandt, Multiscale scientific computation, Six year research summary (1999).
 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).
 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). CrossRef
 F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods (Spinger, New York, 1991).
 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.
 P.M. de Zeeuw, Matrixdependent prolongations and restrictions in a black box multigrid solver, J. Comput. Appl. Math. 33 (1990) 1–27. CrossRef
 J.E. Dendy, Black box multigrid, J. Comput. Phys. 48 (1982) 366–386. CrossRef
 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). CrossRef
 J. Mandel, M. Brezina and P. Vanĕk, Energy optimization of algebraic multigrid bases, Computing 62(3) (1999) 205–228. CrossRef
 J. Ruge and K. Stüben, Algebraic multigrid, in: Multigrid Methods, ed. S. McCormick (SIAM, Philadelphia, PA, 1987).
 K. Stüben, A review of algebraic multigrid, GMD Report 69 (1999).
 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. CrossRef
 P. Vanek, M. Brezina and J. Mandel, Convergence of algebraic multigrid based on smoothed aggregation, Numer. Math. 88(3) (2001) 559–579. CrossRef
 W.L. Wan, T.F. Chan and B. Smith, An energyminimizing interpolation for robust multigrid methods, SIAM J. Sci. Comput. 21(3) (2000) 559–579.
 J. Xu, Iterative methods by space decomposition and subspace correction, SIAM Rev. 34 (1992) 581–613. CrossRef
 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. CrossRef
 Title
 An algebraic multigrid method for finite element systems on crisscross grids
 Journal

Advances in Computational Mathematics
Volume 25, Issue 13 , pp 287304
 Cover Date
 20060701
 DOI
 10.1007/s104440047627y
 Print ISSN
 10197168
 Online ISSN
 15729044
 Publisher
 Kluwer Academic Publishers
 Additional Links
 Topics
 Keywords

 algebraic multigrid method
 finite element
 crisscross grids
 convergence analysis
 Authors

 Shi Shu ^{(001)}
 Jinchao Xu ^{(002)}
 Ying Yang ^{(003)}
 Haiyuan Yu ^{(001)}
 Author Affiliations

 001. Institute for Computational and Applied Mathematics of Xiangtan University, China
 002. Department of Mathematics and Center for Computational Mathematics and Application of Pennsylvania State University, USA
 003. Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and System Sciences, Chinese Academy of Sciences, China