Skip to main content
SpringerLink
Log in
Book cover

International Conference on Parallel Processing and Applied Mathematics

PPAM 2013: Parallel Processing and Applied Mathematics pp 469–478Cite as

Parallel Preconditioner for the Finite Volume Element Discretization of Elliptic Problems

  • Conference paper
  • First Online:

  • 1324 Accesses

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 8385))

Abstract

In this paper we present a parallel preconditioner for the standard Finite Volume (FV) discretization of elliptic problems, using the standard continuous piecewise linear Finite Element (FE) function space. The proposed preconditioner is constructed using an abstract framework of the Additive Schwarz Method, and is fully parallel. The convergence rate of the Generalized Minimal Residual (GMRES) method with this preconditioner is shown to be almost optimal, i.e., it depends poly-logarithmically on the mesh sizes.

This work was partially supported by Polish Scientific Grant 2011/01/B/ST1/01179.

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

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   54.99
Price excludes VAT (USA)
  • Compact, lightweight 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

Learn about institutional subscriptions

References

  1. Lin, Y., Liu, J., Yang, M.: Finite volume element methods: an overview on recent developments. Int. J. Num. Anal. Mod. 4(1), 14–34 (2013)

    MathSciNet  Google Scholar 

  2. Cai, X.C., Widlund, O.B.: Domain decomposition algorithms for indefinite elliptic problems. SIAM J. Sci. Statist. Comput. 13(1), 243–258 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  3. Smith, B.F., Bjørstad, P.E., Gropp, W.D.: Domain Decomposition: Parallel Multilevel Methods For Elliptic Partial Differential Equations. Cambridge University Press, Cambridge (1996)

    MATH  Google Scholar 

  4. Toselli, A., Widlund, O.: Domain Decomposition Methods-Algorithms and Theory. Springer Series in Computational Mathematics, vol. 34. Springer, Berlin (2005)

    MATH  Google Scholar 

  5. Dryja, M., Widlund, O.B.: Schwarz methods of Neumann-Neumann type for three-dimensional elliptic finite element problems. Comm. Pure Appl. Math. 48(2), 121–155 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  6. Brenner, S.C., Wang, K.: Two-level additive Schwarz preconditioners for \(C^0\) interior penalty methods. Numer. Math. 102(2), 231–255 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  7. Mandel, J., Brezina, M.: Balancing domain decomposition for problems with large jumps in coefficients. Math. Comp. 65(216), 1387–1401 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  8. Griebel, M., Oswald, P.: On the abstract theory of additive and multiplicative Schwarz algorithms. Numer. Math. 70(2), 163–180 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  9. Marcinkowski, L.: A balancing Neumann-Neumann method for a mortar finite element discretization of a fourth order elliptic problem. J. Numer. Math. 18(3), 219–234 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  10. Chou, S.H., Huang, J.: A domain decomposition algorithm for general covolume methods for elliptic problems. J. Numer. Math. 11(3), 179–194 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  11. Zhang, S.: On domain decomposition algorithms for covolume methods for elliptic problems. Comput. Meth. Appl. Mech. Engrg. 196(1–3), 24–32 (2006)

    Article  MATH  Google Scholar 

  12. Eisenstat, S.C., Elman, H.C., Schultz, M.H.: Variational iterative methods for nonsymmetric systems of linear equations. SIAM J. Numer. Anal. 20(2), 345–357 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  13. Saad, Y., Schultz, M.H.: GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Statist. Comput. 7(3), 856–869 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  14. Xu, J., Cai, X.C.: A preconditioned GMRES method for nonsymmetric or indefinite problems. Math. Comp. 59(200), 311–319 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  15. Huang, J., Xi, S.: On the finite volume element method for general self-adjoint elliptic problems. SIAM J. Numer. Anal. 35(5), 1762–1774 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  16. Ewing, R.E., Lin, T., Lin, Y.: On the accuracy of the finite volume element method based on piecewise linear polynomials. SIAM J. Numer. Anal. 39(6), 1865–1888 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  17. Cai, X.C.: Some domain decomposition algorithms for nonselfadjoin elliptic and parabolic partial differential equations. Ph.D. thesis, Courant Institute, New York (1989)

    Google Scholar 

  18. Brenner, S.C.: The condition number of the Schur complement in domain decomposition. Numer. Math. 83(2), 187–203 (1999)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Faculty of Mathematics, University of Warsaw, Banacha 2, 02-097, Warszawa, Poland

    Leszek Marcinkowski

  2. Faculty of Engineering, Bergen University College, Nygȧrdsgaten 112, 5020, Bergen, Norway

    Talal Rahman

Authors
  1. Leszek Marcinkowski
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Talal Rahman
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Leszek Marcinkowski .

Editor information

Editors and Affiliations

  1. Institute of Computer and Information Science, Czestochowa University of Technology, Czestochowa, Poland

    Roman Wyrzykowski

  2. University of Tennessee, Department of Computer Science, Knoxville, Tennessee, USA

    Jack Dongarra

  3. Institute of Computer and Information Science, Czestochowa University of Technology, Czestochowa, Poland

    Konrad Karczewski

  4. Technical University of Denmark Informatics and Mathematical Modelling, Kongens Lyngby, Denmark

    Jerzy Waśniewski

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Marcinkowski, L., Rahman, T. (2014). Parallel Preconditioner for the Finite Volume Element Discretization of Elliptic Problems. In: Wyrzykowski, R., Dongarra, J., Karczewski, K., Waśniewski, J. (eds) Parallel Processing and Applied Mathematics. PPAM 2013. Lecture Notes in Computer Science(), vol 8385. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-55195-6_44

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-55195-6_44

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-55194-9

  • Online ISBN: 978-3-642-55195-6

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation