Advertisement

Auxiliary Space Multigrid Method for Elliptic Problems with Highly Varying Coefficients

  • Johannes KrausEmail author
  • Maria Lymbery
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 104)

Abstract

The robust preconditioning of linear systems of algebraic equations arising from discretizations of partial differential equations (PDE) is a fastly developing area of scientific research. In many applications these systems are very large, sparse and therefore it is vital to construct (quasi-)optimal iterative methods that converge independently of problem parameters.

Keywords

Domain Decomposition Method Partial Differential Equation Global Stiffness Matrix Auxiliary Space Subspace Correction 
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.

References

  1. 1.
    O. Axelsson, P. Vassilevski, Algebraic multilevel preconditioning methods I. Numer. Math. 56(2–3), 157–177 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    O. Axelsson, P. Vassilevski, Algebraic multilevel preconditioning methods II. SIAM J. Numer. Anal. 27(6), 1569–1590 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    O. Axelsson, P. Vassilevski, Variable-step multilevel preconditioning methods, I: Self-adjoint and positive definite elliptic problems. Numer. Linear Algebra Appl. 1, 75–101 (1994)MathSciNetzbMATHGoogle Scholar
  4. 4.
    C.R. Dohrmann, A preconditioner for substructuring based on constrained energy minimization. SIAM J. Sci. Comput. 25(1), 246–258 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Y. Efendiev, J. Galvis, R. Lazarov, J. Willems, Robust domain decomposition preconditioners for abstract symmetric positive definite bilinear forms. ESAIM Math. Model. Numer. Anal. 46(05), 1175–1199 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    C. Farhat, F.X. Roux, A method of finite element tearing and interconnecting and its parallel solution algorithm. Int. J. Numer. Methods Eng. 32, 1205–1227 (1991)CrossRefzbMATHGoogle Scholar
  7. 7.
    C. Farhat, M. Lesoinne, P. LeTallec, K. Pierson, D. Rixen, FETI-DP: A dual-primal unified FETI method. I. A faster alternative to the two-level FETI method. Int. J. Numer. Methods Eng. 50(7), 1523–1544 (2001)MathSciNetzbMATHGoogle Scholar
  8. 8.
    J. Galvis, Y. Efendiev, Domain decomposition preconditioners for multiscale flows in high-contrast media. Multiscale Model. Simul. 8(4), 1461–1483 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    I.G. Graham, P.O. Lechner, R. Scheichl, Domain decomposition for multiscale PDEs. Numer. Math. 106(4), 489–626 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    W. Hackbusch, Multi-Grid Methods and Applications (Springer, Berlin, 2003)zbMATHGoogle Scholar
  11. 11.
    X. Hu, P. Vassilevski, J. Xu, Comparative convergence analysis of nonlinear AMLI-cycle multigrid. SIAM J. Numer. Anal. 51(2), 1349–1369 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    A. Klawonn, O. Widlund, M. Dryja, Dual-primal FETI methods for three-dimensional elliptic problems with heterogeneous coefficients. SIAM J. Numer. Anal. 40(1), 159–179 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    J. Kraus, An algebraic preconditioning method for M-matrices: linear versus non-linear multilevel iteration. Numer. Linear Algebra Appl. 9, 599–618 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    J. Kraus, Algebraic multilevel preconditioning of finite element matrices using local Schur complements. Numer. Linear Algebra Appl. 13, 49–70 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    J. Kraus, Additive Schur complement approximation and application to multilevel preconditioning. SIAM J. Sci. Comput. 34, A2872–A2895 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    J. Kraus, P. Vassilevski, L. Zikatanov, Polynomial of best uniform approximation to 1/x and smoothing for two-level methods. Comput. Methods Appl. Math. 12, 448–468 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    J. Kraus, M. Lymbery, S. Margenov, Auxiliary space multigrid method based on additive Schur complement approximation. Numer. Linear Algebra Appl. (2014). doi:10.1002/nla.1959. Online (wileyonlinelibrary.com)Google Scholar
  18. 18.
    Y. Kuznetsov, Algebraic multigrid domain decomposition methods. Sov. J. Numer. Anal. Math. Model. 4(5), 351–379 (1989)MathSciNetzbMATHGoogle Scholar
  19. 19.
    J. Mandel, Balancing domain decomposition. Commun. Numer. Methods Eng. 9(3), 233–241 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    J. Mandel, C.R. Dohrmann, Convergence of a balancing domain decomposition by constraints and energy minimization. Numer. Linear Algebra Appl. 10(7), 639–659 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    J. Mandel, B. Sousedík, Adaptive selection of face coarse degrees of freedom in the BDDC and FETI-DP iterative substructuring methods. Comput. Methods Appl. Mech. Eng. 196(8), 1389–1399 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    J. Mandel, C.R. Dohrmann, R. Tezaur, An algebraic theory for primal and dual substructuring methods by constraints. Appl. Numer. Math. 54(2), 167–193 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    T.P.A. Mathew, Domain Decomposition Methods for the Numerical Solution of Partial Differential Equations (Springer, Berlin, 2008)CrossRefzbMATHGoogle Scholar
  24. 24.
    Y. Notay, P. Vassilevski, Recursive Krylov-based multigrid cycles. Numer. Linear Algebra Appl. 15, 473–487 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    C. Pechstein, R. Scheichl, Analysis of FETI methods for multiscale PDEs. Numer. Math. 111(2), 293–333 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    M. Sarkis, Schwarz preconditioners for elliptic problems with discontinuous coefficients using conforming and non-conforming elements. PhD thesis, Courant Institute, New York University (1994)Google Scholar
  27. 27.
    N. Spillane, V. Dolean, P. Hauret, F. Nataf, C. Pechstein, R. Scheichl, Abstract robust coarse spaces for systems of PDEs via generalized eigenproblems in the overlaps. Numer. Math. 126(4), 741–770 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    A. Toselli, O. Widlund, Domain Decomposition Methods–Algorithms and Theory (Springer, Berlin, 2005)zbMATHGoogle Scholar
  29. 29.
    U. Trottenberg, C.W. Oosterlee, A. Schüller, Multigrid (Academic, San Diego, 2001)zbMATHGoogle Scholar
  30. 30.
    P. Vassilevski, Multilevel Block Factorization Preconditioners: Matrix-based Analysis and Algorithms for Solving Finite Element Equations (Springer, New York, 2008)zbMATHGoogle Scholar
  31. 31.
    J. Xu, The auxiliary space method and optimal multigrid preconditioning techniques for unstructured grids. Computing 56, 215–235 (1996)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.RICAMLinzAustria
  2. 2.IICT, Bulgarian Academy of SciencesSofiaBulgaria

Personalised recommendations