Multigrid Algorithms for High Order Discontinuous Galerkin Methods

  • Paola F. AntoniettiEmail author
  • Marco Sarti
  • Marco Verani
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 104)


In this paper we study the performance of h- and p-multigrid algorithms for high order Discontinuous Galerkin discretizations of elliptic problems. We test the performance of the multigrid schemes employing a wide class of smoothers and considering both two- and three-dimensional test cases.


  1. 1.
    P.F. Antonietti, P. Houston, A class of domain decomposition preconditioners for hp-discontinuous Galerkin finite element methods. J. Sci. Comput. 46(1), 124–149 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    P.F. Antonietti, M. Sarti, M. Verani, Multigrid algorithms for hp-discontinuous Galerkin discretizations of elliptic problems. SIAM J. Numer. Anal. 53(1), 598–618 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    P.F. Antonietti, B. Ayuso, S. Bertoluzza, M. Penacchio, Substructuring preconditioners for an hp domain decomposition method with interior penalty mortaring. Calcolo 52(3), 289–316 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    D.N. Arnold, An interior penalty finite element method with discontinuous elements. SIAM J. Numer. Anal. 19(4), 742–760 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    D.N. Arnold, F. Brezzi, B. Cockburn, L.D. Marini, Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39(5), 1749–1779 (2001/2002)Google Scholar
  6. 6.
    F. Bassi, A. Ghidoni, S. Rebay, P. Tesini, High-order accurate p-multigrid discontinuous Galerkin solution of the Euler equations. Int. J. Numer. Methods Fluids 60(8), 847–865 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    J. Bramble, Multigrid Methods. Number 294 in Pitman Research Notes in Mathematics Series (Longman Scientific & Technical, London, 1993)Google Scholar
  8. 8.
    S.C. Brenner, Convergence of the multigrid V -cycle algorithm for second-order boundary value problems without full elliptic regularity. Math. Comput. 71(238), 507–525 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    S.C. Brenner, Convergence of nonconforming V-cycle and F-cycle multigrid algorithms for second order elliptic boundary value problems. Math. Comput. 73(247), 1041–1066 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    S.C. Brenner, J. Zhao, Convergence of multigrid algorithms for interior penalty methods. Appl. Numer. Anal. Comput. Math. 2(1), 3–18 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    K. Brix, M. Campos Pinto, C. Canuto, W. Dahmen, Multilevel preconditioning of discontinuous Galerkin spectral element methods. Part I: geometrically conforming meshes. IMA J. Numer. Anal. (2014). doi:10.1093/imanum/dru053Google Scholar
  12. 12.
    C. Canuto, L.F. Pavarino, A.B. Pieri, BDDC preconditioners for continuous and discontinuous Galerkin methods using spectral/hp elements with variable local polynomial degree. IMA J. Numer. Anal. 34(3), 879–903 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    B. Cockburn, C.-W. Shu, The local discontinuous Galerkin method for time-dependent convection-diffusion systems. SIAM J. Numer. Anal. 35(6), 2440–2463 (electronic) (1998)Google Scholar
  14. 14.
    K.J. Fidkowski, T.A. Oliver, J. Lu, D.L. Darmofal, p-multigrid solution of high-order discontinuous Galerkin discretizations of the compressible Navier-Stokes equations. J. Comput. Phys. 207(1), 92–113 (2005)Google Scholar
  15. 15.
    E.H. Georgoulis, E. Süli, Optimal error estimates for the hp-version interior penalty discontinuous Galerkin finite element method. IMA J. Numer. Anal. 25(1), 205–220 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    J. Gopalakrishnan, G. Kanschat, A multilevel discontinuous Galerkin method. Numer. Math. 95(3), 527–550 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    W. Hackbusch, Multi-Grid Methods and Applications. Springer Series in Computational Mathematics, vol. 4 (Springer, Berlin, 1985)Google Scholar
  18. 18.
    P. Houston, C. Schwab, E. Süli, Discontinuous hp-finite element methods for advection-diffusion-reaction problems. SIAM J. Numer. Anal. 39(6), 2133–2163 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    J. Kraus, P. Vassilevski, L. Zikatanov, Polynomial of best uniform approximation to 1∕x and smoothing in two-level methods. Comput. Methods Appl. Math. 12(4), 448–468 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    H. Luo, J.D. Baum, R. Löhner, A p-multigrid discontinuous Galerkin method for the Euler equations on unstructured grids. J. Comput. Phys. 211(2), 767–783 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    B.S. Mascarenhas, B.T. Helenbrook, H.L. Atkins, Coupling p-multigrid to geometric multigrid for discontinuous Galerkin formulations of the convection-diffusion equation. J. Comput. Phys. 229(10), 3664–3674 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    C.R. Nastase, D.J. Mavriplis, High-order discontinuous Galerkin methods using an hp-multigrid approach. J. Comput. Phys. 213(1), 330–357 (2006)CrossRefzbMATHGoogle Scholar
  23. 23.
    I. Perugia, D. Schötzau, An hp-analysis of the local discontinuous Galerkin method for diffusion problems. J. Sci. Comput. 17(1–4), 561–571 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    K. Shahbazi, D.J. Mavriplis, N.K. Burgess, Multigrid algorithms for high-order discontinuous Galerkin discretizations of the compressible Navier-Stokes equations. J. Comput. Phys. 228(21), 7917–7940 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    B. Stamm, T.P. Wihler, hp-optimal discontinuous Galerkin methods for linear elliptic problems. Math. Comput. 79(272), 2117–2133 (2010)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Paola F. Antonietti
    • 1
    Email author
  • Marco Sarti
    • 1
  • Marco Verani
    • 1
  1. 1.MOX, Dipartimento di MatematicaPolitecnico di MilanoMilanoItaly

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