Numerische Mathematik

, Volume 119, Issue 1, pp 21–47 | Cite as

Multigrid algorithms for symmetric discontinuous Galerkin methods on graded meshes



We study a class of symmetric discontinuous Galerkin methods on graded meshes. Optimal order error estimates are derived in both the energy norm and the L 2 norm, and we establish the uniform convergence of V-cycle, F-cycle and W-cycle multigrid algorithms for the resulting discrete problems. Numerical results that confirm the theoretical results are also presented.

Mathematics Subject Classification (2000)

65N30 5N15 65N55 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Adams R.A., Fournier J.J.F.: Sobolev Spaces, 2nd edn. Academic Press, Amsterdam (2003)MATHGoogle Scholar
  2. 2.
    Apel Th., Sändig A.-M., Whiteman J.R.: Graded mesh refinement and error estimates for finite element solutions of elliptic boundary value problems in non-smooth domains. Math. Methods Appl. Sci. 19, 63–85 (1996)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Arnold D.N.: An interior penalty finite element method with discontinuous elements. SIAM J. Numer. Anal. 19, 742–760 (1982)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Arnold D.N., Brezzi F., Cockburn B., Marini L.D.: Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39, 1749–1779 (2002)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Babuška I.: Finite element method for domains with corners. Computing 6, 264–273 (1970)MATHCrossRefGoogle Scholar
  6. 6.
    Babuška I., Kellogg R.B., Pitkäranta J.: Direct and inverse error estimates for finite elements with mesh refinements. Numer. Math. 33, 447–471 (1979)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Bank R.E., Dupont T.F.: An optimal order process for solving finite element equations. Math. Comput. 36, 35–51 (1981)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Bassi, F., Rebay, S., Mariotti, G., Pedinotti, S., Savini, M.: A high-order accurate discontinuous finite element method for inviscid and viscous turbomachinery flows. In: Decuypere, R., Dibelius, G. (eds.) Proceedings of 2nd European Conference on Turbomachinery, Fluid Dynamics and Thermodynamics, pp. 99–108. Technologisch Institut, Antwerpen, Belgium (1997)Google Scholar
  9. 9.
    Bramble J.H., Zhang X.: The analysis of multigrid methods. In: Ciarlet, P.G., Lions, J.L. (eds) Handbook of Numerical Analysis, VII, pp. 173–415. North-Holland, Amsterdam (2000)Google Scholar
  10. 10.
    Brannick J.J., Li H., Zikatanov L.T.: Uniform convergence of the multigrid V-cycle on graded meshes for corner singularities. Numer. Linear Algebra Appl. 15, 291–306 (2008)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Brenner S.C.: Convergence of the multigrid V-cycle algorithm for second order boundary value problems without full elliptic regularity. Math. Comput. 71, 507–525 (2002)MathSciNetMATHGoogle Scholar
  12. 12.
    Brenner S.C.: Convergence of nonconforming V-cycle and F-cycle multigrid algorithms for second order elliptic boundary value problems. Math. Comput. 73, 1041–1066 (2004)MathSciNetMATHGoogle Scholar
  13. 13.
    Brenner S.C., Carstensen C.: Finite element methods. In: Stein, E., de Borst, R., Hughes, T.J.R. (eds) Encyclopedia of Computational Mechanics, pp. 73–118. Wiley, Weinheim (2004)Google Scholar
  14. 14.
    Brenner S.C., Cui J., Li F., Sung L.-Y.: A nonconforming finite element method for a two-dimensional curl-curl and grad-div problem. Numer. Math. 109, 509–533 (2008)MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Brenner S.C., Cui J., Sung L.-Y.: Multigrid methods for the symmetric interior penalty method on graded meshes. Numer. Linear Algebra Appl. 16, 481–501 (2009)MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Brenner S.C., Li F., Sung L.-Y.: A locally divergence-free nonconforming finite element method for the time-harmonic Maxwell equations. Math. Comput. 76, 573–595 (2007)MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Brenner S.C., Li F., Sung L.-Y.: A locally divergence-free interior penalty method for two dimensional curl-curl problems. SIAM J. Numer. Anal. 46, 1190–1211 (2008)MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Brenner S.C., Li F., Sung L.-Y.: A nonconforming penalty method for a two dimensional curl-curl problem. M3AS 19, 651–668 (2009)MathSciNetMATHGoogle Scholar
  19. 19.
    Brenner S.C., Owens L.: A W-cycle algorithm for a weakly over-penalized interior penalty method. Comput. Methods Appl. Mech. Eng. 196, 3823–3832 (2007)MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Brenner S.C., Scott L.R.: The Mathematical Theory of Finite Element Methods, 3rd edn. Springer, New York (2008)MATHCrossRefGoogle Scholar
  21. 21.
    Brenner S.C., Sung L.-Y.: Multigrid algorithms for C 0 interior penalty methods. SIAM J. Numer. Anal. 44, 199–223 (2006)MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Brenner S.C., Zhao J.: Convergence of multigrid algorithms for interior penalty methods. Appl. Numer. Anal. Comput. Math. 2, 3–18 (2005)MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    Brezzi F., Manzini G., Marini D., Pietra P., Russo A.: Discontinuous Galerkin approximations for elliptic problems. Numer. Methods Partial Differ. Equ. 16, 365–378 (2000)MathSciNetMATHCrossRefGoogle Scholar
  24. 24.
    Castillo P.: Performance of discontinuous Galerkin methods for elliptic PDEs. SIAM J. Sci. Comput. 24, 524–547 (2002)MathSciNetMATHCrossRefGoogle Scholar
  25. 25.
    Castillo P., Cockburn B., Perugia I., Schötzau D.: An a priori error analysis of the local discontinuous Galerkin method for elliptic problems. SIAM J. Numer. Anal. 38, 1676–1706 (2000)MathSciNetMATHCrossRefGoogle Scholar
  26. 26.
    Cockburn B., Shu C.-W.: The local discontinuous Galerkin method for time-dependent convection-diffusion systems. SIAM J. Numer. Anal. 35, 2440–2463 (1998)MathSciNetMATHCrossRefGoogle Scholar
  27. 27.
    Dauge M.: Elliptic boundary value problems on corner domains. In: Lecture Notes in Mathematics, vol 1341. Springer, Berlin (1988)Google Scholar
  28. 28.
    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, 753–770 (2006)MathSciNetMATHCrossRefGoogle Scholar
  29. 29.
    Douglas, J. Jr., Dupont, T.: Interior penalty procedures for elliptic and parabolic Galerkin methods. In: Computing Methods in Applied Sciences (Second Internat. Sympos., Versailles, 1975), pp. 207–216. Lecture Notes in Phys., vol. 58. Springer, Berlin (1976)Google Scholar
  30. 30.
    Gopalakrishnan J., Kanschat G.: A multilevel discontinuous Galerkin method. Numer. Math. 95, 527–550 (2003)MathSciNetMATHCrossRefGoogle Scholar
  31. 31.
    Grisvard P.: Elliptic Problems in Non Smooth Domains. Pitman, Boston (1985)Google Scholar
  32. 32.
    Gudi T.: A new error analysis for discontinuous finite element methods for linear elliptic problems. Math. Comput. 79, 2169–2189 (2010)MathSciNetMATHCrossRefGoogle Scholar
  33. 33.
    Hackbusch W.: Multi-Grid Methods and Applications. Springer, Berlin-Heidelberg-New York-Tokyo (1985)MATHGoogle Scholar
  34. 34.
    Kondratiev V.: Boundary value problems for elliptic equations in domains with conical or angular points. Trans. Moscow Math. Soc. 16, 227–313 (1967)Google Scholar
  35. 35.
    McCormick, S.F. (eds): Multigrid methods. Frontiers in Applied Mathematics, vol. 3. SIAM, Philadelphia (1987)Google Scholar
  36. 36.
    Nazarov S.A., Plamenevsky B.A.: Elliptic Problems in Domains with Piecewise Smooth Boundaries. de Gruyter, Berlin-New York (1994)MATHCrossRefGoogle Scholar
  37. 37.
    Trottenberg U., Oosterlee C., Schüller A.: Multigrid. Academic Press, San Diego (2001)MATHGoogle Scholar
  38. 38.
    van Raalte M.H., Hemker P.W.: Two-level multigrid analysis for the convection-diffusion equation discretized by a discontinuous Galerkin method. Numer. Linear Algebra Appl. 12, 563–584 (2005)MathSciNetMATHCrossRefGoogle Scholar
  39. 39.
    Wheeler M.F.: An elliptic collocation-finite-element method with interior penalties. SIAM J. Numer. Anal. 15, 152–161 (1978)MathSciNetMATHCrossRefGoogle Scholar
  40. 40.
    Yosida K.: Functional Analysis. Classics in Mathematics. Springer, Berlin (1995) Reprint of the sixth (1980) editionGoogle Scholar
  41. 41.
    Yserentant H.: The convergence of multilevel methods for solving finite-element equations in the presence of singularities. Math. Comput. 47, 399–409 (1986)MathSciNetMATHGoogle Scholar
  42. 42.
    Zhao J.: Convergence of nonconforming V-cycle and F-cycle methods for the biharmonic problem using the Morley element. Electron. Trans. Numer. Anal. 17, 112–132 (2004)MathSciNetMATHGoogle Scholar

Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  • S. C. Brenner
    • 1
    • 2
  • J. Cui
    • 1
    • 3
  • T. Gudi
    • 2
    • 4
  • L.-Y. Sung
    • 1
  1. 1.Department of MathematicsLouisiana State UniversityBaton RougeUSA
  2. 2.Center for Computation and TechnologyLouisiana State UniversityBaton RougeUSA
  3. 3.Institute for Mathematics and its ApplicationsUniversity of MinnesotaMinneapolisUSA
  4. 4.Department of MathematicsIndian Institute of ScienceBangaloreIndia

Personalised recommendations