Skip to main content

Auxiliary Space Multigrid Method for Elliptic Problems with Highly Varying Coefficients

  • Conference paper
Domain Decomposition Methods in Science and Engineering XXII

Part of the book series: Lecture Notes in Computational Science and Engineering ((LNCSE,volume 104))

  • 1464 Accesses

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.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. O. Axelsson, P. Vassilevski, Algebraic multilevel preconditioning methods I. Numer. Math. 56(2–3), 157–177 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  2. O. Axelsson, P. Vassilevski, Algebraic multilevel preconditioning methods II. SIAM J. Numer. Anal. 27(6), 1569–1590 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  4. C.R. Dohrmann, A preconditioner for substructuring based on constrained energy minimization. SIAM J. Sci. Comput. 25(1), 246–258 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  8. J. Galvis, Y. Efendiev, Domain decomposition preconditioners for multiscale flows in high-contrast media. Multiscale Model. Simul. 8(4), 1461–1483 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  9. I.G. Graham, P.O. Lechner, R. Scheichl, Domain decomposition for multiscale PDEs. Numer. Math. 106(4), 489–626 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  10. W. Hackbusch, Multi-Grid Methods and Applications (Springer, Berlin, 2003)

    MATH  Google Scholar 

  11. X. Hu, P. Vassilevski, J. Xu, Comparative convergence analysis of nonlinear AMLI-cycle multigrid. SIAM J. Numer. Anal. 51(2), 1349–1369 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  13. J. Kraus, An algebraic preconditioning method for M-matrices: linear versus non-linear multilevel iteration. Numer. Linear Algebra Appl. 9, 599–618 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  14. J. Kraus, Algebraic multilevel preconditioning of finite element matrices using local Schur complements. Numer. Linear Algebra Appl. 13, 49–70 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  15. J. Kraus, Additive Schur complement approximation and application to multilevel preconditioning. SIAM J. Sci. Comput. 34, A2872–A2895 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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. Y. Kuznetsov, Algebraic multigrid domain decomposition methods. Sov. J. Numer. Anal. Math. Model. 4(5), 351–379 (1989)

    MathSciNet  MATH  Google Scholar 

  19. J. Mandel, Balancing domain decomposition. Commun. Numer. Methods Eng. 9(3), 233–241 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  23. T.P.A. Mathew, Domain Decomposition Methods for the Numerical Solution of Partial Differential Equations (Springer, Berlin, 2008)

    Book  MATH  Google Scholar 

  24. Y. Notay, P. Vassilevski, Recursive Krylov-based multigrid cycles. Numer. Linear Algebra Appl. 15, 473–487 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  25. C. Pechstein, R. Scheichl, Analysis of FETI methods for multiscale PDEs. Numer. Math. 111(2), 293–333 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  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. 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)

    Article  MathSciNet  MATH  Google Scholar 

  28. A. Toselli, O. Widlund, Domain Decomposition Methods–Algorithms and Theory (Springer, Berlin, 2005)

    MATH  Google Scholar 

  29. U. Trottenberg, C.W. Oosterlee, A. Schüller, Multigrid (Academic, San Diego, 2001)

    MATH  Google Scholar 

  30. P. Vassilevski, Multilevel Block Factorization Preconditioners: Matrix-based Analysis and Algorithms for Solving Finite Element Equations (Springer, New York, 2008)

    MATH  Google Scholar 

  31. J. Xu, The auxiliary space method and optimal multigrid preconditioning techniques for unstructured grids. Computing 56, 215–235 (1996)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Johannes Kraus .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer International Publishing Switzerland

About this paper

Cite this paper

Kraus, J., Lymbery, M. (2016). Auxiliary Space Multigrid Method for Elliptic Problems with Highly Varying Coefficients. In: Dickopf, T., Gander, M., Halpern, L., Krause, R., Pavarino, L. (eds) Domain Decomposition Methods in Science and Engineering XXII. Lecture Notes in Computational Science and Engineering, vol 104. Springer, Cham. https://doi.org/10.1007/978-3-319-18827-0_3

Download citation

Publish with us

Policies and ethics