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Algebraic Multigrid Based on Computational Molecules, 1: Scalar Elliptic Problems

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Abstract

We consider the problem of splitting a symmetric positive definite (SPD) stiffness matrix A arising from finite element discretization into a sum of edge matrices thereby assuming that A is given as the sum of symmetric positive semidefinite (SPSD) element matrices. We give necessary and sufficient conditions for the existence of an exact splitting into SPSD edge matrices and address the problem of best positive (nonnegative) approximation.

Based on this disassembling process we present a new concept of ``strong'' and ``weak'' connections (edges), which provides a basis for selecting the coarse-grid nodes in algebraic multigrid methods. Furthermore, we examine the utilization of computational molecules (small collections of edge matrices) for deriving interpolation rules. The reproduction of edge matrices on coarse levels offers the opportunity to combine classical coarsening algorithms with effective (energy minimizing) interpolation principles yielding a flexible and robust new variant of AMG.

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References

  • Axelsson, O.: Iterative solution methods. Cambridge University Press 1994.

  • A. Brandt (1983) Algebraic multigrid theory: the symmetric case. Preliminary Proc. Int. Multigrid Conf Copper Mountain CO

    Google Scholar 

  • A Brandt (1986) ArticleTitleAlgebraic multigrid theory: the symmetric case Appl. Math. Comput. 19 23–56 Occurrence Handle10.1016/0096-3003(86)90095-0 Occurrence Handle0616.65037 Occurrence Handle87j:65042

    Article  MATH  MathSciNet  Google Scholar 

  • Brandt, A., McCormick, S. F., Ruge, J. W.: Algebraic multigrid (AMG) for automatic multigrid solutions with applications to geodetic computations. Report, Inst. for Computational Studies, Fort Collins, CO, 1982.

  • A. Brandt S. F. McCormick J. W. Ruge (1985) Algebraic multigrid (AMG) for sparse matrix equations D. J. Evans (Eds) Sparsity and its applications Cambridge University Press Cambridge 257–284

    Google Scholar 

  • M. Brezina A. J. Cleary R. D. Falgout V. E. Henson J. E. Jones T. A. Manteuffel S. F. McCormick J. W. Ruge (2000) ArticleTitleAlgebraic multigrid based on element interpolation (AMGe) SIAM J. Sci. Comput. 22 1570–1592 Occurrence Handle10.1137/S1064827598344303 Occurrence Handle2001k:65184

    Article  MathSciNet  Google Scholar 

  • Briggs, W. L., Henson, V. E., McCormick, S. F.: A multigrid tutorial, 2nd edition. Philadelphia: SIAM Books 2001.

  • Chan, T. F., Vanek, P.: Detection of strong coupling in algebraic multigrid solvers. In: Multigrid methods VI, vol. 14. Berlin: Springer 2000, pp. 11–23.

  • T. Chartier R. D. Falgout V. E. Henson J. Jones T. Manteuffel S. McCormick J. Ruge P. S. Vassilevski (2004) ArticleTitleSpectral AMGe (ρAMGe) SIAM J. Sci. Comput. 25 1–26 Occurrence Handle2004m:65036

    MathSciNet  Google Scholar 

  • A. J. Cleary R. D. Falgout V. E. Henson J. E. Jones T. A. Manteuffel S. F. McCormick G. N. Miranda J. W. Ruge (2000) ArticleTitleRobustness and scalability of algebraic multigrid SIAM J. Sci. Stat. Comput. 21 1886–1908 Occurrence Handle10.1137/S1064827598339402 Occurrence Handle2001f:65043

    Article  MathSciNet  Google Scholar 

  • R. D. Falgout P. S. Vassilevski (2004) ArticleTitleOn generalizing the algebraic multigrid framework SIAM J. Numer. Anal. 42 1669–1693 Occurrence Handle10.1137/S0036142903429742 Occurrence Handle2005i:65047

    Article  MathSciNet  Google Scholar 

  • G. Haase U. Langer S. Reitzinger J. Schöberl (2004) ArticleTitleAlgebraic multigrid methods based on element preconditioning Int. J. Comp. Math. 78 575–598

    Google Scholar 

  • V. E. Henson P. Vassilevski (2001) ArticleTitleElement-free AMGe: general algorithms for computing the interpolation weights in AMG SIAM J. Sci. Comput. 23 629–650 Occurrence Handle10.1137/S1064827500372997 Occurrence Handle2002j:65040

    Article  MathSciNet  Google Scholar 

  • J. E. Jones P. Vassilevski (2001) ArticleTitleAMGe based on element agglomeration SIAM J. Sci. Comput. 23 109–133 Occurrence Handle10.1137/S1064827599361047 Occurrence Handle2002g:65035

    Article  MathSciNet  Google Scholar 

  • Kraus, J. K.: On the utilization of edge matrices in algebraic multigrid. Lecture Notes in Computer Science. Proc. 5th Int. Conf. on ``Large-Scale Scientific Computations'', Sozopol, Bulgaria, June 6–10, 2005 (to appear).

  • Langer, U., Reitzinger, S., Schicho, J.: Symbolic methods for the element preconditioning technique. In: Proc. SNSC Hagenberg (Langer, U., Winkler, F., eds.). Springer 2002.

  • J. Mandel M. Brezina P. Vaněk (1999) ArticleTitleEnergy optimization of algebraic multigrid bases Computing 62 205–228 Occurrence Handle10.1007/s006070050022 Occurrence Handle2000j:65124

    Article  MathSciNet  Google Scholar 

  • J. W. Ruge K. Stüben (1985) Efficient solution of finite difference and finite element equations by algebraic multigrid (AMG). Multigrid methods for integral and differential equations D. J. Paddon H. Holstein (Eds) The Institute of Mathematics and Its Applications Conference Series Clarendon Press Oxford 169–212

    Google Scholar 

  • Ruge, J. W., Stüben, K.: Algebraic multigrid (AMG). In: Multigrid methods. Frontiers in Applied Mathematics, Vol. 3 (McCormick, S.F., ed.). Philadelphia: SIAM 1987, pp. 73–130.

  • Schöberl, J.: NETGEN – An advancing front 2D/3D-mesh generator based on abstract rules. Comput. Visual. Sci. 41–52 (1997).

  • K. Stüben (1983) ArticleTitleAlgebraic multigrid (AMG): experiences and comparisons Appl. Math. Comput. 13 419–452 Occurrence Handle10.1016/0096-3003(83)90023-1 Occurrence Handle0533.65064 Occurrence Handle85a:65069

    Article  MATH  MathSciNet  Google Scholar 

  • Tuminaro, R. S., Tong, C.: Parallel smoothed aggregation multigrid: aggregation strategies on massively parallel machines. Report, Sandia National Laboratories, 2000.

  • Varga, R. S.: Matrix iterative analysis. Prentice-Hall 1962.

  • P. Vaněk M. Brezina J. Mandel (2001) ArticleTitleConvergence of algebraic multigrid based on smoothed aggregation Numer. Math. 88 559–579 Occurrence Handle2002c:65230

    MathSciNet  Google Scholar 

  • P. Vaněk J. Mandel M. Brezina (1996) ArticleTitleAlgebraic multigrid based on smoothed aggregation for second- and fourth-order problems Computing 56 179–196 Occurrence Handle97c:65207

    MathSciNet  Google Scholar 

  • W. L. Wan T. F. Chan B. Smith (2000) ArticleTitleAn energy-minimizing interpolation for robust multigrid methods Siam J. Sci. Comput. 21 1632–1649 Occurrence Handle2001a:65162

    MathSciNet  Google Scholar 

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  1. Johann Radon Institute for Computational and Applied Mathematics, Altenbergerstraße 69, 4040, Linz, Austria

    J. K. Kraus & J. Schicho

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  1. J. K. Kraus
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  2. J. Schicho
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Kraus, J.K., Schicho, J. Algebraic Multigrid Based on Computational Molecules, 1: Scalar Elliptic Problems. Computing 77, 57–75 (2006). https://doi.org/10.1007/s00607-005-0147-x

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