Skip to main content
SpringerLink
Log in

A Class of Domain Decomposition Preconditioners for hp-Discontinuous Galerkin Finite Element Methods

  • Published:
Journal of Scientific Computing Aims and scope Submit manuscript

Abstract

In this article we address the question of efficiently solving the algebraic linear system of equations arising from the discretization of a symmetric, elliptic boundary value problem using hp-version discontinuous Galerkin finite element methods. In particular, we introduce a class of domain decomposition preconditioners based on the Schwarz framework, and prove bounds on the condition number of the resulting iteration operators. Numerical results confirming the theoretical estimates are also presented.

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

Similar content being viewed by others

References

  1. Adams, R.A.: Sobolev Spaces. Pure and Applied Mathematics, vol. 65. Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York (1975)

    MATH  Google Scholar 

  2. Ainsworth, M.: A preconditioner based on domain decomposition for h-p finite-element approximation on quasi-uniform meshes. SIAM J. Numer. Anal. 33(4), 1358–1376 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  3. 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 

  4. Antonietti, P.F., Ayuso, B.: Class of preconditioners for discontinuous Galerkin approximations of elliptic problems. In: Domain Decomposition Methods in Science and Engineering XVII. Lect. Notes Comput. Sci. Eng., vol. 60, pp. 185–192. Springer, Berlin (2008)

    Chapter  Google Scholar 

  5. Antonietti, P.F., Ayuso, B.: Multiplicative Schwarz methods for discontinuous Galerkin approximations of elliptic problems. M2AN Math. Model. Numer. Anal. 42(3), 443–469 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  6. Antonietti, P.F., Ayuso, B.: Two-level Schwarz preconditioners for super penalty discontinuous Galerkin methods. Commun. Comput. Phys. 5(2–4), 398–412 (2009)

    MathSciNet  Google Scholar 

  7. 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 

  8. Arnold, D.N., Brezzi, F., Cockburn, B., Marini, L.D.: Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39(5), 1749–1779 (2001/02) (electronic)

    Article  MathSciNet  Google Scholar 

  9. Babuška, I., Craig, A., Mandel, J., Pitkäranta, J.: Efficient preconditioning for the p-version finite element method in two dimensions. SIAM J. Numer. Anal. 28(3), 624–661 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  10. Bjørstad, P.E.: Multiplicative and additive Schwarz methods: convergence in the two-domain case. In: Domain Decomposition Methods, Los Angeles, CA, 1988, pp. 147–159. SIAM, Philadelphia (1989)

    Google Scholar 

  11. Brenner, S.C.: Poincaré-Friedrichs inequalities for piecewise H 1 functions. SIAM J. Numer. Anal. 41(1), 306–324 (2003) (electronic)

    Article  MATH  MathSciNet  Google Scholar 

  12. 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 

  13. Brezzi, F., Manzini, G., Marini, D., Pietra, P., Russo, A.: Discontinuous finite elements for diffusion problems. In: Atti Convegno in onore di F. Brioschi (Milano 1997), pp. 197–217. Istituto Lombardo, Accademia di Scienze e Lettere (1999)

  14. Brezzi, F., Manzini, G., Marini, D., Pietra, P., Russo, A.: Discontinuous Galerkin approximations for elliptic problems. Numer. Methods Partial Differ. Equ. 16(4), 365–378 (2000)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  16. Cai, X.-C., Widlund, O.B.: Multiplicative Schwarz algorithms for some nonsymmetric and indefinite problems. SIAM J. Numer. Anal. 30(4), 936–952 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  17. Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. Studies in Mathematics and its Applications, vol. 4. North-Holland, Amsterdam (1978)

    Book  MATH  Google Scholar 

  18. 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) (electronic)

    Article  MATH  MathSciNet  Google Scholar 

  19. Dawson, C., Sun, S., Wheeler, M.F.: Compatible algorithms for coupled flow and transport. Comput. Methods Appl. Mech. Eng. 193(23–26), 2565–2580 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  20. Dohrmann, C.R., Klawonn, A., Widlund, O.B.: Domain decomposition for less regular subdomains: Overlapping Schwarz in two dimensions. SIAM J. Numer. Anal. 46(4), 2153–2168 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  21. 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 

  22. Dubiner, M.: Spectral methods on triangles and other domains. J. Sci. Comput. 6(4), 345–390 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  23. 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 

  24. 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) (electronic)

    Article  MATH  MathSciNet  Google Scholar 

  25. Georgoulis, E.H., Hall, E., Houston, P.: Discontinuous Galerkin methods on hp-anisotropic meshes. I. A priori error analysis. Int. J. Comput. Sci. Math. 1(2–4), 221–244 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  26. Georgoulis, E.H., Hall, E., Houston, P.: Discontinuous Galerkin methods for advection-diffusion-reaction problems on anisotropically refined meshes. SIAM J. Sci. Comput. 30(1), 246–271 (2007/08)

    Article  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  28. Guo, B., Cao, W.: A preconditioner for the h-p version of the finite element method in two dimensions. Numer. Math. 75(1), 59–77 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  29. Guo, B., Cao, W.: An iterative and parallel solver based on domain decomposition for the h-p version of the finite element method. J. Comput. Appl. Math. 83(1), 71–85 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  30. Guo, B., Cao, W.: An additive Schwarz method for the h-p version of the finite element method in three dimensions. SIAM J. Numer. Anal. 35(2), 632–654 (1998) (electronic)

    Article  MATH  MathSciNet  Google Scholar 

  31. Heinrich, B., Pietsch, K.: Nitsche type mortaring for some elliptic problem with corner singularities. Computing 68(3), 217–238 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  32. Hesthaven, J.S., Warburton, T.: Nodal Discontinuous Galerkin Methods. Texts in Applied Mathematics, vol. 54. Springer, New York (2008). Algorithms, analysis, and applications

    Book  MATH  Google Scholar 

  33. Houston, P., Schwab, C., Süli, E.: Discontinuous hp-finite element methods for advection-diffusion-reaction problems. SIAM J. Numer. Anal. 39(6), 2133–2163 (2002) (electronic)

    Article  MATH  MathSciNet  Google Scholar 

  34. Lasser, C., Toselli, A.: An overlapping domain decomposition preconditioner for a class of discontinuous Galerkin approximations of advection-diffusion problems. Math. Comput. 72(243), 1215–1238 (2003) (electronic)

    Article  MATH  MathSciNet  Google Scholar 

  35. Pavarino, L.F.: Additive Schwarz methods for the p-version finite element method. Numer. Math. 66(4), 493–515 (1994)

    MATH  MathSciNet  Google Scholar 

  36. Pavarino, L.F.: Schwarz methods with local refinement for the p-version finite element method. Numer. Math. 69(2), 185–211 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  37. Pavarino, L.F., Widlund, O.B.: A polylogarithmic bound for an iterative substructuring method for spectral elements in three dimensions. SIAM J. Numer. Anal. 33(4), 1303–1335 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  38. Pavarino, L.F., Widlund, O.B.: Iterative substructuring methods for spectral elements: problems in three dimensions based on numerical quadrature. Comput. Math. Appl. 33(1–2), 193–209 (1997). Approximation theory and applications

    Article  MATH  MathSciNet  Google Scholar 

  39. Quarteroni, A., Valli, A.: Numerical Approximation of Partial Differential Equations. Springer Series in Computational Mathematics, vol. 23. Springer, Berlin (1994)

    MATH  Google Scholar 

  40. Rivière, B.: Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations. Theory and Implementation. Frontiers in Applied Mathematics, vol. 35. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2008)

    MATH  Google Scholar 

  41. Rivière, B., Wheeler, M.F., Girault, V.: Improved energy estimates for interior penalty, constrained and discontinuous Galerkin methods for elliptic problems. I. Comput. Geosci. 3(3–4), 337–360 (2000)

    Article  Google Scholar 

  42. Sherwin, S.J., Karniadakis, G.E.: A new triangular and tetrahedral basis for high-order (hp) finite element methods. Int. J. Numer. Methods Eng. 38(22), 3775–3802 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  43. 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 

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

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. MOX–Modeling and Scientific Computing, Dipartimento di Matematica, Politecnico di Milano, Piazza Leondardo da Vinci 32, 20133, Milano, Italy

    Paola F. Antonietti

  2. School of Mathematical Sciences, University of Nottingham, University Park, Nottingham, NG7 2RD, UK

    Paul Houston

Authors
  1. Paola F. Antonietti
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Paul Houston
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Paola F. Antonietti.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Antonietti, P.F., Houston, P. A Class of Domain Decomposition Preconditioners for hp-Discontinuous Galerkin Finite Element Methods. J Sci Comput 46, 124–149 (2011). https://doi.org/10.1007/s10915-010-9390-1

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10915-010-9390-1

Keywords

Search

Navigation