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Numerische Mathematik

, Volume 132, Issue 1, pp 23–49 | Cite as

A robust multigrid method for discontinuous Galerkin discretizations of Stokes and linear elasticity equations

  • Qingguo Hong
  • Johannes KrausEmail author
  • Jinchao Xu
  • Ludmil Zikatanov
Article

Abstract

We consider multigrid methods for discontinuous Galerkin \(H({\text {div}},\Omega )\)-conforming discretizations of the Stokes and linear elasticity equations. We analyze variable V-cycle and W-cycle multigrid methods with nonnested bilinear forms. We prove that these algorithms are optimal and robust, i.e., their convergence rates are independent of the mesh size and also of the material parameters such as the Poisson ratio. Numerical experiments are conducted that further confirm the theoretical results.

Mathematics Subject Classification

65N55 65N30 

References

  1. 1.
    Cockburn, B., Kanschat, G., Schötzau, D.: A note on discontinuous Galerkin divergence-free solutions of the Navier–Stokes equations. J. Sci. Comput. 31(1), 61–73 (2007)zbMATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Chen, G., Li, D., Schötzau, D., Wei, X.: A mixed finite element method with exactly divergence-free velocities for incompressible magnetohydrodynamics. Comput. Methods Appl. Mech. Eng. 199(45), 2840–2855 (2010)zbMATHGoogle Scholar
  3. 3.
    Elman, H.C., Silvester, D.J., Wathen, A.J.: Finite Elements and Fast Iterative Solvers: with Applications in Incompressible Fluid Dynamics. Numerical Mathematics and Scientific Computation. Oxford University Press, New York (2005)Google Scholar
  4. 4.
    Ayuso, B., Brezzi, F., Marini, L.D., Xu, J., Zikatanov, L.: A simple preconditioner for a discontinuous Galerkin method for the Stokes problem. arXiv preprint (2012). arXiv:1209.5223
  5. 5.
    Xu, J.: The auxiliary space method and optimal multigrid preconditioning techniques for unstructured grids. Computing 56(3), 215–235 (1996)zbMATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Nepomnyaschikh, S.V.: Mesh theorems on traces, normalizations of function traces and their inversion. Soviet J. Numer. Anal. Math. Model. 6(3), 223–242 (1991)zbMATHMathSciNetGoogle Scholar
  7. 7.
    Oosterlee, C.W., Lorenz, F.J.: Multigrid methods for the Stokes system. Comput. Sci. Eng. 8(6), 34–43 (2006)CrossRefGoogle Scholar
  8. 8.
    Vanka, S.P.: Block-implicit multigrid solution of Navier–Stokes equations in primitive variables. J. Comput. Phys. 65(1), 138–158 (1986)zbMATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Fortin, M., Glowinski, R.: Augmented Lagrangian Methods: Applications to the Numerical Solution of Boundary Value Problems. Translated from the French by B. Hunt and D. C. Spicer. Studies in Mathematics and its Applications, vol. 15. North-Holland Publishing Co., Amsterdam (1983)Google Scholar
  10. 10.
    Schöberl, Joachim: Multigrid methods for a parameter dependent problem in primal variables. Numer. Math. 84(1), 97–119 (1999)zbMATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    Karer, E., Kraus, J., Zikatanov, L.: A subspace correction method for nearly singular elasticity problems. In: Bank, R., et al. (eds.) Domain Decomposition Methods in Science and Engineering XX. Lecture Notes in Computational Science and Engineering, vol. 91, pp. 165–172. Springer, Berlin Heidelberg (2013)Google Scholar
  12. 12.
    Lee, Y.J., Wu, J., Xu, L., Zikatanov, J.: Robust subspace correction methods for nearly singular systems. Math. Models Methods Appl. Sci. 17(11), 1937–1963 (2007)zbMATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    Arnold, D.N., Falk, R.S., Winther, R.: Multigrid in H(div) and H(curl). Numer. Math. 85(2), 197–217 (2000)zbMATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    Hackbusch, W.: Multigrid Methods and Applications, vol. 4. Springer, Berlin (1985)CrossRefGoogle Scholar
  15. 15.
    Bramble, J.H., Pasciak, J.E., Xu, J.: The analysis of multigrid algorithms with nonnested spaces or noninherited quadratic forms. Math. Comput. 56(193), 1–34 (1991)zbMATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    Girault, V., Raviart, P.A.: Finite Element Methods for Navier-Stokes Equations: Theory and Algorithms, vol. 87. Springer, Berlin (1986)zbMATHCrossRefGoogle Scholar
  17. 17.
    Bramble, James H.: A proof of the inf-sup condition for the Stokes equations on Lipschitz domains. Math. Models Methods Appl. Sci. 13(3), 361–371 (2003)zbMATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    Kellogg, R.B., Osborn, J.E.: A regularity result for the Stokes problem in a convex polygon. J. Funct. Anal. 21(4), 397–431 (1976)zbMATHMathSciNetCrossRefGoogle Scholar
  19. 19.
    Brenner, S.C., Sung, L.Y.: Linear finite element methods for planar linear elasticity. Math. Comput. 59, 321–321 (1992)zbMATHMathSciNetCrossRefGoogle Scholar
  20. 20.
    Brenner, S.C., Scott, R.: The Mathematical Theory of Finite Element Methods, 3rd edn. Texts in Applied Mathematics, vol. 15. Springer, New York (2007)Google Scholar
  21. 21.
    Langer, U., Queck, W.: On the convergence factor of Uzawa’s algorithm. J. Comput. Appl. Math. 15(2), 191–202 (1986)zbMATHMathSciNetCrossRefGoogle Scholar
  22. 22.
    Brenner, S.C.: Korn’s inequalities for piecewise \({H}^1\) vector fields. Math. Comput. 73, 1067–1088 (2004)zbMATHCrossRefGoogle Scholar
  23. 23.
    Schötzau, D., Schwab, C., Toselli, A.: Mixed hp-DGFEM for incompressible flows. SIAM J. Numer. Anal. 40(6), 2171–2194 (2002)MathSciNetCrossRefGoogle Scholar
  24. 24.
    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)zbMATHMathSciNetCrossRefGoogle Scholar
  25. 25.
    Monk, P.: Finite Element Methods for Maxwell’s Equations. Oxford University Press, USA (2003)zbMATHCrossRefGoogle Scholar
  26. 26.
    Boffi, D., Brezzi, F., Fortin, M.: Mixed and Hybrid Finite Element Methods. Springer Series in Computational Mathematics, vol. 44. Springer, New York (2013)Google Scholar
  27. 27.
    Babuška, I., Aziz, A.K.: Lectures on the mathematical foundations of the finite element method. University of Maryland, College Park, Washington DC. Technical Note BN-748 (1972)Google Scholar
  28. 28.
    Babuška, I.: The finite element method with Lagrangian multipliers. Numer. Math. 20:179–192, (1972/73)Google Scholar
  29. 29.
    Xu, J.: Iterative methods by space decomposition and subspace correction. SIAM Rev. 34, 581–613 (1992)zbMATHMathSciNetCrossRefGoogle Scholar
  30. 30.
    Xu, J., Zikatanov, L.: The method of alternating projections and the method of subspace corrections in Hilbert space. J. Am. Math. Soc. 15(3), 573–598 (2002)zbMATHMathSciNetCrossRefGoogle Scholar
  31. 31.
    Bergh, J., Löfström, J.: Interpolation Spaces: an Introduction. Springer, Berlin (1976). Grundlehren der Mathematischen Wissenschaften, No. 223zbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Qingguo Hong
    • 1
  • Johannes Kraus
    • 2
    Email author
  • Jinchao Xu
    • 3
  • Ludmil Zikatanov
    • 3
    • 4
  1. 1.Johann Radon InstituteAustrian Academy of SciencesLinzAustria
  2. 2.Faculty of MathematicsUniversity of Duisburg-EssenEssenGermany
  3. 3.Department of MathematicsPennsylvania State UniversityUniversity ParkUSA
  4. 4.Institute of Mathematics and InformaticsBulgarian Academy of SciencesSofiaBulgaria

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