Robust Multigrid for Cartesian Interior Penalty DG Formulations of the Poisson Equation in 3D

  • Jörg StillerEmail author
Conference paper
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 119)


We present a polynomial multigrid method for the nodal interior penalty formulation of the Poisson equation on three-dimensional Cartesian grids. Its key ingredient is a weighted overlapping Schwarz smoother operating on element-centered subdomains. The MG method reaches superior convergence rates corresponding to residual reductions of about two orders of magnitude within a single V(1,1) cycle. It is robust with respect to the mesh size and the ansatz order, at least up to P = 32. Rigorous exploitation of tensor-product factorization yields a computational complexity of O(PN) for N unknowns, whereas numerical experiments indicate even linear runtime scaling. Moreover, by allowing adjustable subdomain overlaps and adding Krylov acceleration, the method proved feasible for anisotropic grids with element aspect ratios up to 48.



Funding by German Research Foundation (DFG) in frame of the project STI 157/4-1 is gratefully acknowledged.


  1. 1.
    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)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    P. Bastian, M. Blatt, R. Scheichl Algebraic multigrid for discontinuous Galerkin discretizations of heterogeneous elliptic problems. Numer. Linear Algebra Appl. 19, 367–388 (2012)Google Scholar
  3. 3.
    J. Bramble, Multigrid Methods. Pitman Research Notes Mathematical Series, vol. 294 (Longman Scientific & Technical, Harlow, 1995)Google Scholar
  4. 4.
    B. Cockburn, G.E. Karniadakis, C.-W. Shu, Discontinuous Galerkin Methods: Theory, Computation and Applications (Springer, Berlin, Heidelberg, 2000)CrossRefzbMATHGoogle Scholar
  5. 5.
    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, 92–113 (2005)Google Scholar
  6. 6.
    G.H. Golub, Q. Ye Inexact preconditioned conjugate gradient method with inner-outer iteration. SIAM J. Sci. Comput. 21(4), 1305–1320 (1999)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    J. Gopalakrishnan, G. Kanschat, A multilevel discontinuous Galerkin method. Numer. Math. 95, 527–550 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    L. Haupt, J. Stiller, W. Nagel, A fast spectral element solver combining static condensation and multigrid techniques. J. Comput. Phys. 255, 384–395 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    B.T. Helenbrook, H.L. Atkins, D.J. Mavriplis, Analysis of p-multigrid for continuous and discontinuous finite element discretizations. AIAA Paper 2003–3989 (2003)Google Scholar
  10. 10.
    B.T. Helenbrook, H.L. Atkins, Application of p-multigrid to discontinuous Galerkin formulations of the poisson equation. AIAA J. 44, 566–575 (2006)CrossRefGoogle Scholar
  11. 11.
    J.S. Hesthaven, T. Warburton, Nodal Discontinuous Galerkin Methods (Springer, Berlin, 2008)CrossRefzbMATHGoogle Scholar
  12. 12.
    E.F. Kaasschieter, Preconditioned conjugate gradients for solving singular systems. J. Comput. Appl. Math. 24, 265–275 (1988)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    G. Kanschat, Multilevel methods for discontinuous Galerkin FEM on locally refined meshes. Comput. Struct. 82, 2437–2445 (2004)CrossRefGoogle Scholar
  14. 14.
    J.K. Kraus, S.K. Tomar, A multilevel method for discontinuous Galerkin approximation of three-dimensional anisotropic elliptic problems. Numer Linear Algebra Appl. 15, 417–438 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    R.E. Lynch, J.R. Rice, D.H. Thomas, Direct solution of partial difference equations by tensor product methods. Numer. Math. 6, 185–199 (1964)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    L.N. Olson, J.B. Schroder, Smoothed aggregation multigrid solvers for high-order discontinuous Galerkin methods for elliptic problems. J. Comput. Phys. 230, 6959–6976 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    B. Rivière, Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations: Theory and Implementation. Frontiers in Mathematics, vol. 35 (SIAM, Philadelphia, PA, 2008)Google Scholar
  18. 18.
    J. Stiller, Robust multigrid for high-order discontinuous Galerkin methods: a fast Poisson solver suitable for high-aspect ratio Cartesian grids. J. Comput. Phys. 327, 317–336 (2016)CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    J. Stiller, Nonuniformly weighted Schwarz smoothers for spectral element multigrid. J. Sci. Comput. 72(1), 81–96 (2017)CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    U. Trottenberg, C.W. Oosterlee, A. Schüller, Multigrid (Academic, New York, 2000)zbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Institute of Fluid Mechanics and Center of Advancing Electronics Dresden (cfaed), TU DresdenDresdenGermany

Personalised recommendations