Skip to main content
SpringerLink
Log in

Space Decompositions and Solvers for Discontinuous Galerkin Methods

  • Conference paper
  • First Online:
Domain Decomposition Methods in Science and Engineering XXI

Part of the book series: Lecture Notes in Computational Science and Engineering ((LNCSE,volume 98))

  • 1285 Accesses

Abstract

We present a brief overview of the different domain and space decomposition techniques that enter in developing and analyzing solvers for discontinuous Galerkin methods. Emphasis is given to the novel and distinct features that arise when considering DG discretizations over conforming methods. Connections and differences with the conforming approaches are emphasized.

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 149.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 199.99
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. Antonietti, P.F., Ayuso, B.: Schwarz domain decomposition preconditioners for discontinuous Galerkin approximations of elliptic problems: non-overlapping case. Math. Model. Numer. Anal. 41(1), 21–54 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  2. Antonietti, P.F., Ayuso, B.: Multiplicative Schwarz methods for discontinuous Galerkin approximations of elliptic problems. Math. Model. Numer. Anal. 42(3), 443–469 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  3. Antonietti, P.F., Ayuso de Dios, B., Bertoluzza, S., Penacchio, M.: Substructuring preconditioners for an h-p Nitsche-type method, Calcolo, doi:10.1007/s10092-014-0117-9. Published online march 2014

    Google Scholar 

  4. Arnold, D.N.: An interior penalty finite element method with discontinuous elements. SIAM J. Numer. Anal. 19(4), 742–760 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  5. Ayuso de Dios, B., Zikatanov, L.: Uniformly convergent iterative methods for discontinuous Galerkin discretizations. J. Sci. Comput. 40(1–3), 4–36 (2009)

    Google Scholar 

  6. Ayuso de Dios, B., Georgiev, I., Kraus, J., Zikatanov, L.: A subspace correction methods for discontinuous Galerkin methods discretizations for linear elasticity equations. ESAIM Math. Model. Numer. Anal. 47(5), 1315–1333 (2013). http://dx.doi.org/10.1051/m2an/2013070

  7. Ayuso de Dios, B., Holst, M., Zhu, Y., Zikatanov, L.: Multilevel preconditioners for discontinuous Galerkin approximations of elliptic problems with jump coefficients. Math. Comput. 83(287), 1083–1120 (2014)

    Google Scholar 

  8. Ayuso de Dios, B., Lombardi, A., Pietra, P., Zikatanov, L.: A block solver for the exponentially fitted IIPG-0 method. In: Twenty International Conference on Domain Decomposition. Lecture Notes in Computational Science and Engineering, vol. 91, pp. 247–255. Springer, Berlin (2013). http://dx.doi.org/10.1007/978-3-642-35275-1_28

  9. Barker, A.T., Brenner, S.C., Park, E.-H., Sung, L.-Y.: Two-level additive Schwarz preconditioners for a weakly over-penalized symmetric interior penalty method. J. Sci. Comput. 47(1), 27–49 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  10. Brenner, S.C.: Two-level additive Schwarz preconditioners for nonconforming finite element methods. Math. Comput. 65(215), 897–921 (1996)

    Article  MATH  Google Scholar 

  11. Brenner, S.C., Zhao, J.: Convergence of multigrid algorithms for interior penalty methods. Appl. Numer. Anal. Comput. Math. 2(1), 3–18 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  12. Brezzi, F., Cockburn, B., Marini, L.D., Süli, E.: Stabilization mechanisms in discontinuous Galerkin finite element methods. Comput. Methods Appl. Mech. Eng. 195(25–28), 3293–3310 (2006)

    Article  MATH  Google Scholar 

  13. Brix, K., Campos Pinto, M., Dahmen, W.: A multilevel preconditioner for the interior penalty discontinuous Galerkin method. SIAM J. Numer. Anal. 46(5), 2742–2768 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  14. Brix, K., Campos Pinto, M., Canuto, C., Dahmen, W.: Multilevel preconditioning of discontinuous Galerkin spectral element methods. Part I: geometrically conforming meshes. Technical Report, IGPM Preprint, RWTH Aachen (2013). ArXiv:1301.6768

    Google Scholar 

  15. Canuto, C., Pavarino, L., Pieri, A.: BDDC preconditioners for continuous and discontinuous Galerkin methods using spectral/hp elements with variable local polynomial degree. IMA J. Numer. Anal. (2013). doi:10.1093/imanum/drt037

    Google Scholar 

  16. Chung, E.T., Kim, H.H., Widlund, O.B.: Two-level overlapping Schwarz algorithms for a staggered discontinuous Galerkin method. SIAM J. Numer. Anal. 51(1), 47–67 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  17. Dobrev, V.A., Lazarov, R.D., Vassilevski, P.S., Zikatanov, L.T.: Two-level preconditioning of discontinuous Galerkin approximations of second-order elliptic equations. Numer. Linear Algebra Appl. 13(9), 753–770 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  18. Dryja, M., Galvis, J., Sarkis, M.: BDDC methods for discontinuous Galerkin discretization of elliptic problems. J. Complex. 23(4–6), 715–739 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  19. Dryja, M., Galvis, J., Sarkis, M.: Neumann-Neumann methods for a DG discretization of elliptic problems with discontinuous coefficients on geometrically nonconforming substructures. Numer. Methods Partial Differ. Equ. 28(4), 1194–1226 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  20. Dryja, M., Sarkis, M.: FETI-DP method for a composite finite element and discontinuous Galerkin method. SIAM J. Numer. Anal. 51(1), 400–422 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  21. 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(4), 1343–1365 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  22. Gopalakrishnan, J., Kanschat, G.: A multilevel discontinuous Galerkin method. Numer. Math. 95(3), 527–550 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  23. Hajian, S., Gander, M.: Block Jacobi for discontinuous Galerkin discretizations: no ordinary Schwarz methods. In: Domain Decomposition Methods in Science and Engineering XXI. Lecture Notes in Computational Science and Engineering. Springer, Berlin (2014) (same volume)

    Google Scholar 

  24. Karakashian, O.A., Pascal, F.: A posteriori error estimates for a discontinuous Galerkin approximation of second-order elliptic problems. SIAM J. Numer. Anal. 41(6), 2374–2399 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  25. Lions, P.L.: On the Schwarz alternating method. I. In: First International Symposium on Domain Decomposition Methods for Partial Differential Equations (Paris, 1987), pp. 1–42. SIAM, Philadelphia (1988)

    Google Scholar 

  26. Nepomnyaschikh, S.V.: Mesh theorems on traces, normalizations of function traces and their inversion. Sov. J. Numer. Anal. Math. Model. 6(3), 223–242 (1991)

    MATH  MathSciNet  Google Scholar 

  27. Oswald, P.: Preconditioners for nonconforming discretizations. Math. Comput. 65(215), 923–941 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  28. Sarkis, M.: Nonstandard coarse spaces and Schwarz methods for elliptic problems with discontinuous coefficients using non-conforming elements. Numer. Math. 77(3), 383–406 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  29. Schöberl, J., Lehrenfeld, C.: Domain decomposition preconditioning for high order hybrid discontinuous Galerkin methods on tetrahedral meshes. In: Advanced Finite Element Methods and Applications. Lecture Notes in Applied and Computational Mechanics, vol. 66, pp. 27–56. Springer, Berlin (2013)

    Google Scholar 

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

    Google Scholar 

  31. Widlund, O.B.: Some Schwarz methods for symmetric and nonsymmetric elliptic problems. In: Fifth International Symposium on Domain Decomposition Methods for Partial Differential Equations. SIAM, Philadelphia (1992)

    Google Scholar 

  32. Xu, J.: Iterative methods by space decomposition and subspace correction. SIAM Rev. 34(4), 581–613 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  33. Xu, J.: The auxiliary space method and optimal multigrid preconditioning techniques for unstructured grids. Computing 56(3), 215–235 (1996). International GAMM-Workshop on Multi-level Methods (Meisdorf, 1994)

    Google Scholar 

  34. Xu, J., Zikatanov, L.: The method of alternating projections and the method of subspace corrections in Hilbert space. J. Am. Math. Soc. 15(3), 573–597 (2002)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgements

Blanca Ayuso de Dios thanks R. Hiptmair (ETH) for raising up the issue on the use of Auxiliary space techniques for preconditioning DG methods, at the DD21 meeting. First author has been partially supported by MINECO grant MTM2011-27739-C04-04 and GENCAT 2009SGR-345.

Author information

Authors and Affiliations

  1. Center for Uncertainty Quantification in Computational Science & Engineering, Division of Mathematics & Computer, Electrical and Mathematical Sciences & Engineering (CEMSE), King Abdullah University of Science and Technology, Thuwal, 23955-6900, Kingdom of Saudi Arabia

    Blanca Ayuso de Dios

  2. Department of Mathematics, The Pennsylvania State University, University Park, PA, USA

    Ludmil Zikatanov

Authors
  1. Blanca Ayuso de Dios
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ludmil Zikatanov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Blanca Ayuso de Dios .

Editor information

Editors and Affiliations

  1. INRIA, Rennes, France

    Jocelyne Erhel

  2. Section de Mathématiques, Université de Genève, Genève, Switzerland

    Martin J. Gander

  3. Laboratoire Analyse, Géometrie & Applications, Université Paris XIII, Villetaneuse, France

    Laurence Halpern

  4. INRIA, Rennes, France

    Géraldine Pichot

  5. Laboratoire LMNO, Université de Caen, Caen, France

    Taoufik Sassi

  6. New York University Courant Institute, New York, New York, USA

    Olof Widlund

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer International Publishing Switzerland

About this paper

Cite this paper

de Dios, B.A., Zikatanov, L. (2014). Space Decompositions and Solvers for Discontinuous Galerkin Methods. In: Erhel, J., Gander, M., Halpern, L., Pichot, G., Sassi, T., Widlund, O. (eds) Domain Decomposition Methods in Science and Engineering XXI. Lecture Notes in Computational Science and Engineering, vol 98. Springer, Cham. https://doi.org/10.1007/978-3-319-05789-7_1

Download citation

Publish with us

Policies and ethics

Search

Navigation