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Additive Average Schwarz with Adaptive Coarse Space for Morley FE

  • Salah AlrabeeiEmail author
  • Mahmood Jokar
  • Leszek Marcinkowski
Conference paper
  • 97 Downloads
Part of the Lecture Notes in Computer Science book series (LNCS, volume 12044)

Abstract

We propose an additive average Schwarz preconditioner with two adaptively enriched coarse space for the nonconforming Morley finite element method for fourth order biharmonic equation with highly varying and discontinuous coefficients. In this paper, we extend the work of [9, 10]: (additive average Schwarz with adaptive coarse spaces: scalable algorithms for multiscale problems). Our analysis shows that the condition number of the preconditioned problem is bounded independent of the jump of the coefficient, and it depends only on the ratio of the coarse to the fine mesh.

Keywords

Additive average Schwarz Nonconforming finite element Domain decomposition methods Fourth order problems with highly varying coefficients 

Notes

Acknowledgments

The authors are deeply thankful for Prof. Talal Rahman for his invaluable comments, discussions, and suggestions in this work.

References

  1. 1.
    Bjørstad, P.E., Dryja, M., Rahman, T.: Additive Schwarz methods for elliptic mortar finite element problems. Numer. Math. 95(3), 427–457 (2003)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Bjørstad, P.E., Dryja, M., Vainikko, E.: Additive Schwarz methods without subdomain overlap and with new coarse spaces. In: 1995 Domain Decomposition Methods in Sciences and Engineering, Beijing, pp. 141–157 (1997)Google Scholar
  3. 3.
    Brenner, S.C.: Two-level additive Schwarz preconditioners for nonconforming finite element methods. Math. Comp. 65(215), 897–921 (1996)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Brenner, S.C., Sung, L.Y.: Balancing domain decomposition for nonconforming plate elements. Numer. Math. 83(1), 25–52 (1999)MathSciNetCrossRefGoogle Scholar
  5. 5.
    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
  6. 6.
    Dryja, M., Sarkis, M.: Additive average Schwarz methods for discretization of elliptic problems with highly discontinuous coefficients. Comput. Methods Appl. Math. 10(2), 164–176 (2010).  https://doi.org/10.2478/cmam-2010-0009MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Feng, X., Rahman, T.: An additive average Schwarz method for the plate bending problem. J. Numer. Math. 10(2), 109–125 (2002)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Larson, M.G., Bengzon, F.: The Finite Element Method: Theory, Implementation, and Applications, vol. 10. Springer, Heidelberg (2013).  https://doi.org/10.1007/978-3-642-33287-6CrossRefzbMATHGoogle Scholar
  9. 9.
    Marcinkowski, L., Rahman, T.: Two new enriched multiscale coarse spaces for the additive average Schwarz method. In: Lee, C.-O., et al. (eds.) Domain Decomposition Methods in Science and Engineering XXIII. LNCSE, vol. 116, pp. 389–396. Springer, Cham (2017).  https://doi.org/10.1007/978-3-319-52389-7_40CrossRefGoogle Scholar
  10. 10.
    Marcinkowski, L., Rahman, T.: Additive average Schwarz with adaptive coarse spaces: scalable algorithms for multiscale problems. Electron. Trans. Numer. Anal. 49, 28–40 (2018).  https://doi.org/10.1553/etna_vol49s28MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Morley, L.S.D.: The triangular equilibrium problem in the solution of plate bending problems. Aero. Quart. 23(19), 149–169 (1968)CrossRefGoogle Scholar
  12. 12.
    Rahman, T., Xu, X., Hoppe, R.: Additive Schwarz methods for the Crouzeix-Raviart mortar finite element for elliptic problems with discontinuous coefficients. Numer. Math. 101(3), 551–572 (2005)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Smith, B., Bjorstad, P., Gropp, W.: Domain Decomposition: Parallel Multilevel Methods for Elliptic Partial Differential Equations. Cambridge University Press, Cambridge (2004)zbMATHGoogle Scholar
  14. 14.
    Xu, X., Lui, S., Rahman, T.: A two-level additive Schwarz method for the morley nonconforming element approximation of a nonlinear biharmonic equation. IMA J. Numer. Anal. 24(1), 97–122 (2004)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  • Salah Alrabeei
    • 1
    Email author
  • Mahmood Jokar
    • 1
  • Leszek Marcinkowski
    • 2
  1. 1.Department of Computing, Mathematics, and PhysicsWestern Norway University of Applied SciencesBergenNorway
  2. 2.Faculty of Mathematics, Informatics, and MechanicsUniversity of WarsawWarszawaPoland

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