Skip to main content
SpringerLink
Log in

A Two-Level Additive Schwarz Preconditioner for Local \(C^0\) Discontinuous Galerkin Methods of Kirchhoff Plates

  • Original Paper
  • Published:
Communications on Applied Mathematics and Computation Aims and scope Submit manuscript

Abstract

A two-level additive Schwarz preconditioner based on the overlapping domain decomposition approach is proposed for the local \(C^0\) discontinuous Galerkin (LCDG) method of Kirchhoff plates. Then with the help of an intergrid transfer operator and its error estimates, it is proved that the condition number is bounded by \(O(1+(H^4/\delta ^4))\), where H is the diameter of the subdomains and \(\delta \) measures the overlap among subdomains. And for some special cases of small overlap, the estimate can be improved as \(O(1+(H^3/\delta ^3))\). At last, some numerical results are reported to demonstrate the high efficiency of the two-level additive Schwarz preconditioner.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1

Similar content being viewed by others

References

  1. Brenner, S.C.: A two-level additive Schwarz preconditioner for macro-element approximations of the plate bending problem. Houst. J. Math. 21, 823–844 (1995)

    MathSciNet  MATH  Google Scholar 

  2. Brenner, S.C.: A two-level additive Schwarz preconditioner for nonconforming plate elements. Numer. Math. 72, 419–447 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  3. Brenner, S.C.: Two-level additive Schwarz preconditioners for nonconforming finite element methods. Math. Comput. 65, 897–921 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  4. Brenner, S.C., Scott, L.R.: The Mathematical Theory of Finite Element Methods, 3rd edn. Springer, New York (2008)

    Book  MATH  Google Scholar 

  5. Brenner, S.C., Sung, L.: \(C^0\) interior penalty methods for fourth order elliptic boundary value problems on polygonal domains. J. Sci. Comput. 22(23), 83–118 (2005)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  7. Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978)

    MATH  Google Scholar 

  8. Cockburn, B., Shu, C.-W.: The local discontinuous Galerkin method for time-dependent convection-diffusion systems. SIAM J. Numer. Anal. 35(6), 2440–2463 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  9. Dryja, M., Widlund, O.B.: Some domain decomposition algorithms for elliptic problems. In: Iterative Methods for Large Linear Systems, pp. 273–291. Academic Press, Boston (1990)

  10. Dryja, M., Widlund, O.B.: Domain decomposition algorithms with small overlap. SIAM J. Sci. Comput. 15, 604–620 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  11. Engel, G., Garikipati, K., Hughes, T.J.R., Larson, M.G., Mazzei, L., Taylor, R.L.: Continuous/discontinuous finite element approximations of fourth-order elliptic problems in structural and continuum mechanics with applications to thin beams and plates, and strain gradient elasticity. Comput. Methods Appl. Mech. Eng. 191, 3669–3750 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  12. Feng, K., Shi, Z.: Mathematical Theory of Elastic Structures. Springer, Berlin (1995)

    Google Scholar 

  13. Feng, X., Karakashian, O.A.: Two-level additive Schwarz methods for a discontinuous Galerkin approximation of second order elliptic problems. SIAM J. Numer. Anal. 39, 1343–1365 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  14. Feng, X., Karakashian, O.A.: Two-level non-overlapping Schwarz preconditioners for a discontinuous Galerkin approximation of the biharmonic equation. J. Sci. Comput. 22(23), 289–314 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  15. Huang, J., Huang, X., Han, W.: A new \(C^0\) discontinuous Galerkin method for Kirchhoff plates. Comput. Methods Appl. Mech. Eng. 199, 1446–1454 (2010)

    Article  MATH  Google Scholar 

  16. Lasser, C., Toselli, A.: An overlapping domain decomposition preconditioner for a class of discontinuous Galerkin approximations of advection-diffusion problems. Math. Comput. 72, 1215–1238 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  17. Nepomnyaschikh, S.V.: On the application of the bordering method to the mixed boundary value problem for elliptic equations and on mesh norms in \(W_{2}^{1/2}(S)\). Sov. J. Numer. Anal. Math. Model. 4, 493–506 (1989)

    MathSciNet  Google Scholar 

  18. Reddy, J.N.: Theory and Analysis of Elastic Plates and Shells, 2nd edn. CRC Press, New York (2006)

    Google Scholar 

  19. Wells, G.N., Dung, N.T.: A \(C^0\) discontinuous Galerkin formulation for Kirchhoff plates. Comput. Methods Appl. Mech. Eng. 196, 3370–3380 (2007)

    Article  MATH  Google Scholar 

  20. Zhang, X.: Two-level Schwarz methods for the biharmonic problem discretized conforming \(C^1\) elements. SIAM J. Numer. Anal. 33, 555–570 (1996)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors thank the referees for their valuable comments leading to improvement of an early version of the paper. The first author was supported under NSFC (Grant no. 11571237). The second author was supported under NSFC (Grant no. 11771338), and the Fundamental Research Funds for the Central Universities.

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, and MOE-LSC, Shanghai Jiao Tong University, Shanghai, 200240, China

    Jianguo Huang

  2. School of Mathematics, Shanghai University of Finance and Economics, Shanghai, 200433, China

    Xuehai Huang

Authors
  1. Jianguo Huang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xuehai Huang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jianguo Huang.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Huang, J., Huang, X. A Two-Level Additive Schwarz Preconditioner for Local \(C^0\) Discontinuous Galerkin Methods of Kirchhoff Plates. Commun. Appl. Math. Comput. 1, 167–185 (2019). https://doi.org/10.1007/s42967-019-0003-1

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s42967-019-0003-1

Keywords

Mathematics Subject Classification

Search

Navigation