Skip to main content
SpringerLink
Log in

Overlapping Schwarz Preconditioner for Fourth Order Multiscale Elliptic Problems

  • Conference paper
  • First Online:
Parallel Processing and Applied Mathematics (PPAM 2019)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 12044))

  • 587 Accesses

Abstract

In this paper, a domain decomposition parallel preconditioner for the 4th order multiscale elliptic problem in 2D with highly heterogeneous coefficients is constructed. The problem is discretized by the conforming \(C^1\) reduced Hsieh-Tough-Tocher (HCT) macro element. The proposed preconditioner is based on the classical overlapping Schwarz method and is constructed using an abstract framework of the Additive Schwarz Method. The coarse space consists of multiscale finite element functions associated with the edges and is enriched with functions based on solving carefully constructed generalized eigenvalue problems locally on each edge. The convergence rate of the Preconditioned Conjugate Method of our method is independent of the variations in the coefficients for a sufficient number of eigenfunctions in the coarse space.

L. Marcinkowski—The work was partially supported by Polish Scientific Grant: National Science Center 2016/21/B/ST1/00350.

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

Access this chapter

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

Institutional subscriptions

References

  1. Brenner, S.C., Scott, L.R.: The Mathematical Theory of Finite Element Methods, Texts in Applied Mathematics, vol. 15, Second edn. Springer, New York (2002). https://doi.org/10.1007/978-1-4757-3658-8

    Book  Google Scholar 

  2. Ciarlet, P.G.: Basic error estimates for elliptic problems. In: Handbook of Numerical Analysis, vol. II, pp. 17–351, North-Holland, Amsterdam (1991)

    Google Scholar 

  3. Efendiev, Y., Galvis, J., Lazarov, R., Willems, J.: Robust domain decomposition preconditioners for abstract symmetric positive definite bilinear forms. ESAIM Math. Model. Numer. Anal. 46(5), 1175–1199 (2012). https://doi.org/10.1051/m2an/2011073

    Article  MathSciNet  MATH  Google Scholar 

  4. Efendiev, Y., Galvis, J., Vassilevski, P.S.: Spectral element agglomerate algebraic multigrid methods for elliptic problems with high-contrast coefficients. In: Huang, Y., Kornhuber, R., Widlund, O., Xu, J. (eds.) Domain Decomposition Methods in Science and Engineering XIX. LNCSE, vol. 78, pp. 407–414. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-11304-8_47

    Chapter  MATH  Google Scholar 

  5. Gander, M.J., Loneland, A., Rahman, T.: Analysis of a new harmonically enriched multiscale coarse space for domain decompostion methods. Eprint arXiv:1512.05285 (2015). https://arxiv.org/abs/1512.05285

  6. Graham, I.G., Lechner, P.O., Scheichl, R.: Domain decomposition for multiscale PDEs. Numer. Math. 106(4), 589–626 (2007). https://doi.org/10.1007/s00211-007-0074-1

    Article  MathSciNet  MATH  Google Scholar 

  7. Hackbusch, W.: Iterative Solution of Large Sparse Systems of Equations, Applied Mathematical Sciences, vol. 95. Springer, New York (1994). https://doi.org/10.1007/978-3-319-28483-5. Translated and revised from the 1991 German original

    Book  MATH  Google Scholar 

  8. Le Tallec, P., Mandel, J., Vidrascu, M.: A Neumann-Neumann domain decomposition algorithm for solving plate and shell problems. SIAM J. Numer. Anal. 35(2), 836–867 (1998)

    Article  MathSciNet  Google Scholar 

  9. Pechstein, C.: Finite and Boundary Element Tearing and Interconnecting Solvers for Multiscale Problems. LNCSE, vol. 90. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-23588-7

    Book  MATH  Google Scholar 

  10. Pechstein, C., Scheichl, R.: Analysis of FETI methods for multiscale PDEs. Numer. Math. 111(2), 293–333 (2008). https://doi.org/10.1007/s00211-008-0186-2

    Article  MathSciNet  MATH  Google Scholar 

  11. Scheichl, R., Vainikko, E.: Additive Schwarz with aggregation-based coarsening for elliptic problems with highly variable coefficients. Computing 80(4), 319–343 (2007). https://doi.org/10.1007/s00607-007-0237-z

    Article  MathSciNet  MATH  Google Scholar 

  12. 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 

  13. Spillane, N., Dolean, V., Hauret, P., Nataf, F., Pechstein, C., Scheichl, R.: Abstract robust coarse spaces for systems of PDEs via generalized eigenproblems in the overlaps. Numer. Math. 126(4), 741–770 (2014). https://doi.org/10.1007/s00211-013-0576-y

    Article  MathSciNet  MATH  Google Scholar 

  14. Toselli, A., Widlund, O.: Domain Decomposition Methods–Algorithms and Theory. SSCM, vol. 34. Springer, Berlin (2005). https://doi.org/10.1007/b137868

    Book  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

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

    Leszek Marcinkowski

  2. Department of Computing, Mathematics, and Physics, Western Norway University of Applied Sciences, Inndalsveien 28, 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. Czestochowa University of Technology, Czestochowa, Poland

    Roman Wyrzykowski

  2. University of Southern California, Marina del Rey, CA, USA

    Ewa Deelman

  3. University of Tennessee, Knoxville, TN, USA

    Jack Dongarra

  4. Czestochowa University of Technology, Czestochowa, Poland

    Konrad Karczewski

Rights and permissions

Reprints and permissions

Copyright information

© 2020 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Marcinkowski, L., Rahman, T. (2020). Overlapping Schwarz Preconditioner for Fourth Order Multiscale Elliptic Problems. In: Wyrzykowski, R., Deelman, E., Dongarra, J., Karczewski, K. (eds) Parallel Processing and Applied Mathematics. PPAM 2019. Lecture Notes in Computer Science(), vol 12044. Springer, Cham. https://doi.org/10.1007/978-3-030-43222-5_21

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-43222-5_21

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-43221-8

  • Online ISBN: 978-3-030-43222-5

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation