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Introduction

  • James LottesEmail author
Chapter
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Part of the Springer Theses book series (Springer Theses)

Abstract

This chapter presents the formulation and discretization (by the finite element method) of two simple model problems, the symmetric Poisson’s equation and the non-symmetric advection-diffusion equation. For the symmetric problem, a brief presentation and analysis of the method of solution by geometric multigrid is included. The purpose is to introduce, in a concrete setting, all of the concepts appearing in later chapters, including the variational setting, linear iterative methods, and multigrid in its general conception. This material is background, and for the most part, later chapters can be read independently of this one, except that the non-symmetric model problem will be used for illustrative purposes throughout.

References

  1. 1.
    Bramble, J.H., Pasciak, J.E., Wang, J., Xu, J.: Convergence estimates for multigrid algorithms without regularity assumptions. Math. Comput. 57(195), 23–45 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Brandt, A.: Algebraic multigrid theory: the symmetric case. Appl. Math. Comput. 19(1–4), 23–56 (1986)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Brandt, A.: General highly accurate algebraic coarsening. Electron. Trans. Num. Anal. 10, 1–20 (2000)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Brandt, A., McCormick, S.F., Ruge, J.W.: Algebraic multigrid (AMG) for automatic multigrid ssolution with application to geodetic computations. Technical report. Institute for Computational Studies, Coloroda State University (1982)Google 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, pp. 257–284. Cambridge University Press (1985)Google Scholar
  6. 6.
    Brannick, J., Brezina, M., MacLachlan, S., Manteuffel, T.A., McCormick, S.F., Ruge, J.W.: An energy-based AMG coarsening strategy. Numer. Linear Algebra Appl. 13, 133–148 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Brooks, A.N., Hughes, T.J.: Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations. Comput. Methods Appl. Mech. Eng. 32(1–3), 199–259 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Elman, H.C., Silvester, D.J., Wathen, A.J.: Finite Elements and Fast Iterative Solvers: With Applications in Incompressible Fluid Dynamics. Oxford University Press (2005)Google Scholar
  9. 9.
    Evans, L.C.: Partial Differential Equations, Graduate Studies in Mathematics, vol. 19. American Mathematical Society (1998)Google Scholar
  10. 10.
    Falgout, R.D., Vassilevski, P.S.: On generalizing the algebraic multigrid framework. SIAM J. Numer. Anal. 42(4), 1669–1693 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Falgout, R.D., Vassilevski, P.S., Zikatanov, L.T.: On two-grid convergence estimates. Numer. Linear Algebra Appl. 12(5–6), 471–494 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Fedorenko, R.P.: A relaxation method for solving elliptic difference equations. USSR Comput. Math. Math. Phys. 1, 1092–1096 (1962)CrossRefzbMATHGoogle Scholar
  13. 13.
    Fedorenko, R.P.: The rate of convergence of an iterative process. USSR Comput. Math. Math. Phys. 4, 227–235 (1964)CrossRefzbMATHGoogle Scholar
  14. 14.
    Fedorenko, R.P.: Iterative methods for elliptic difference equations. Russ. Math. Surv. 28, 129–195 (1973)CrossRefzbMATHGoogle Scholar
  15. 15.
    Golub, G.H., Van Loan, C.F.: Matrix Computations, 3rd edn. Johns Hopkins (1996)Google Scholar
  16. 16.
    Hackbusch, W.: Multi-Grid Methods and Applications. Springer, Springer Series in Computational Mathematics (1985)Google Scholar
  17. 17.
    Jones, J.E., Vassilevski, P.S.: AMGe based on element agglomeration. SIAM J. Sci. Comput. 23(1), 109–133 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Kim, H.H., Xu, J., Zikatanov, L.: A multigrid method based on graph matching for convectiondiffusion equations. Numer. Linear Algebra Appl. 10(1–2), 181–195 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Mandel, J., McCormick, S.F., Ruge, J.W.: An algebraic theory for multigrid methods for variational problems. SIAM J. Numer. Anal. 25(1), 91–110 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Notay, Y.: Algebraic analysis of two-grid methods: The nonsymmetric case. Numer. Linear Algebra Appl. 17, 73–96 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Notay, Y.: Aggregation-based algebraic multigrid for convection-diffusion equations. SIAM J. Sci. Comput. 34(4), A2288–A2316 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Quarteroni, A., Valli, A.: Numerical Approximation of Partial Differential Equations. Springer, Springer Series in Computational Mathematics (1994)zbMATHGoogle Scholar
  23. 23.
    Ruge, J.W., Stüben, K.: Algebraic multigrid. In: S.F. McCormick (ed.) Multigrid Methods, pp. 73–130. SIAM (1987)Google Scholar
  24. 24.
    Saad, Y.: Iterative Methods for Sparse Linear Systems, 2 edn. SIAM (2003)Google Scholar
  25. 25.
    Stüben, K.: Algebraic multigrid (AMG): An introduction with applications. In: Trottenberg, U., Oosterlee, C.W., Schüller, A. (eds.) Multigrid. Academic Press (2001)Google Scholar
  26. 26.
    Vaněk, P.: Acceleration of convergence of a two-level algorithm by smoothing transfer operators. Appl. Math. 37(4), 265–274 (1992)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Vassilevski, P.S.: Multilevel Block Factorization Preconditioners: Matrix-based Analysis and Algorithms for Solving Finite Element Equations. Springer (2008)Google Scholar
  28. 28.
    Wan, W.L., Chan, T.F., Smith, B.: An energy-minimizing interpolation for robust multigrid methods. SIAM J. Sci. Comput. 21(4), 1632–1649 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Xu, J., Zikatanov, L.: The method of alternating projections and the method of subspace corrections in Hilbert space. J. Am. Math. Soc. 15(3), 573–597 (2002)MathSciNetCrossRefzbMATHGoogle Scholar

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© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Google Inc.Mountain ViewUSA

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