Journal of Scientific Computing

, Volume 64, Issue 1, pp 1–34 | Cite as

Algebraic and Discretization Error Estimation by Equilibrated Fluxes for Discontinuous Galerkin Methods on Nonmatching Grids

  • Vít Dolejší
  • Ivana Šebestová
  • Martin Vohralík
Article

Abstract

We derive a posteriori error estimates for the discontinuous Galerkin method applied to the Poisson equation. We allow for a variable polynomial degree and simplicial meshes with hanging nodes and propose an approach allowing for simple (nonconforming) flux reconstructions in such a setting. We take into account the algebraic error stemming from the inexact solution of the associated linear systems and propose local stopping criteria for iterative algebraic solvers. An algebraic error flux reconstruction is introduced in this respect. Guaranteed reliability and local efficiency are proven. We next propose an adaptive strategy combining both adaptive mesh refinement and adaptive stopping criteria. At last, we detail a form of the estimates avoiding any practical reconstruction of a flux and only working with the approximate solution, which simplifies greatly their evaluation. Numerical experiments illustrate a tight control of the overall error, good prediction of the distribution of both the discretization and algebraic error components, and efficiency of the adaptive strategy.

Keywords

Linear diffusion problems Discontinuous Galerkin method A posteriori error estimate Flux reconstruction  Stopping criteria Algebraic error 

References

  1. 1.
    Ainsworth, M.: Robust a posteriori error estimation for nonconforming finite element approximation. SIAM J. Numer. Anal. 42(6), 2320–2341 (2005)MATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Ainsworth, M., Rankin, R.: Fully computable error bounds for discontinuous Galerkin finite element approximations on meshes with an arbitrary number of levels of hanging nodes. SIAM J. Numer. Anal. 47(6), 4112–4141 (2010)MATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Ainsworth, M., Rankin, R.: Constant free error bounds for nonuniform order discontinuous Galerkin finite-element approximation on locally refined meshes with hanging nodes. IMA J. Numer. Anal. 31(1), 254–280 (2011)MATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Arioli, M.: A stopping criterion for the conjugate gradient algorithm in a finite element method framework. Numer. Math. 97(1), 1–24 (2004)MATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Arioli, M., Georgoulis, E.H., Loghin, D.: Stopping criteria for adaptive finite element solvers. SIAM J. Sci. Comput. 35(3), A1537–A1559 (2013)MATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Arioli, M., Liesen, J., Międlar, A., Strakoš, Z.: Interplay between discretization and algebraic computation in adaptive numerical solution of elliptic PDE problems. GAMM-Mitt. 36(1), 102–129 (2013)MATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Arioli, M., Loghin, D.: Stopping criteria for mixed finite element problems. Electron. Trans. Numer. Anal. 29, 178–192 (2007/08)Google Scholar
  8. 8.
    Arioli, M., Loghin, D., Wathen, A.J.: Stopping criteria for iterations in finite element methods. Numer. Math. 99(3), 381–410 (2005)MATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Becker, R., Johnson, C., Rannacher, R.: Adaptive error control for multigrid finite element methods. Computing 55(4), 271–288 (1995)MATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Brezzi, F., Fortin, M.: Mixed and Hybrid Finite Element Methods, Springer Series in Computational Mathematics, vol. 15. Springer, New York (1991)CrossRefGoogle Scholar
  11. 11.
    Cheddadi, I., Fučík, R., Prieto, M.I., Vohralík, M.: Computable a posteriori error estimates in the finite element method based on its local conservativity: improvements using local minimization. ESAIM Proc. 24, 77–96 (2008)MATHCrossRefGoogle Scholar
  12. 12.
    Cochez-Dhondt, S., Nicaise, S.: Equilibrated error estimators for discontinuous Galerkin methods. Numer. Method. Part. Differ. Equ. 24(5), 1236–1252 (2008)Google Scholar
  13. 13.
    Di Pietro, D.A., Ern, A.: Mathematical Aspects of Discontinuous Galerkin Methods, Mathématiques & Applications (Berlin) [Mathematics & Applications], vol. 69. Springer, Heidelberg (2012)Google Scholar
  14. 14.
    Drkošová, J., Greenbaum, A., Rozložník, M., Strakoš, Z.: Numerical stability of GMRES. BIT 35(3), 309–330 (1995)MATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    Ern, A., Stephansen, A.F., Vohralík, M.: Guaranteed and robust discontinuous Galerkin a posteriori error estimates for convection–diffusion–reaction problems. J. Comput. Appl. Math. 234(1), 114–130 (2010)MATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    Ern, A., Vohralík, M.: Flux reconstruction and a posteriori error estimation for discontinuous Galerkin methods on general nonmatching grids. C. R. Math. Acad. Sci. Paris 347(7–8), 441–444 (2009)MATHMathSciNetCrossRefGoogle Scholar
  17. 17.
    Ern, A., Vohralík, M.: Adaptive inexact Newton methods with a posteriori stopping criteria for nonlinear diffusion PDEs. SIAM J. Sci. Comput. 35(4), A1761–A1791 (2013)MATHCrossRefGoogle Scholar
  18. 18.
    Greenbaum, A., Rozložník, M., Strakoš, Z.: Numerical behaviour of the modified Gram–Schmidt GMRES implementation. BIT 37(3), 706–719 (1997)MATHMathSciNetCrossRefGoogle Scholar
  19. 19.
    Jiránek, P., Strakoš, Z., Vohralík, M.: A posteriori error estimates including algebraic error and stopping criteria for iterative solvers. SIAM J. Sci. Comput. 32(3), 1567–1590 (2010)MATHMathSciNetCrossRefGoogle Scholar
  20. 20.
    Karakashian, O.A., Pascal, F.: A posteriori error estimates for a discontinuous Galerkin approximation of second-order elliptic problems. SIAM J. Numer. Anal. 41(6), 2374–2399 (2003)MATHMathSciNetCrossRefGoogle Scholar
  21. 21.
    Kim, K.Y.: A posteriori error analysis for locally conservative mixed methods. Math. Comput. 76(257), 43–66 (2007)MATHCrossRefGoogle Scholar
  22. 22.
    Liesen, J., Strakoš, Z.: Krylov Subspace Methods: Principles and Analysis. Numerical Mathematics and Scientific Computation. Oxford University Press, Oxford (2013)Google Scholar
  23. 23.
    Meidner, D., Rannacher, R., Vihharev, J.: Goal-oriented error control of the iterative solution of finite element equations. J. Numer. Math. 17(2), 143–172 (2009)MATHMathSciNetCrossRefGoogle Scholar
  24. 24.
    Nicaise, S.: A posteriori error estimations of some cell-centered finite volume methods. SIAM J. Numer. Anal. 43(4), 1481–1503 (2005)MATHMathSciNetCrossRefGoogle Scholar
  25. 25.
    Paige, C.C., Rozložník, M., Strakoš, Z.: Modified Gram–Schmidt (MGS), least squares, and backward stability of MGS–GMRES. SIAM J. Matrix Anal. Appl. 28(1), 264–284 (2006)MATHMathSciNetCrossRefGoogle Scholar
  26. 26.
    Payne, L.E., Weinberger, H.F.: An optimal Poincaré inequality for convex domains. Arch. Ration. Mech. Anal. 5, 286–292 (1960)MATHMathSciNetCrossRefGoogle Scholar
  27. 27.
    Pencheva, G.V., Vohralík, M., Wheeler, M.F., Wildey, T.: Robust a posteriori error control and adaptivity for multiscale, multinumerics, and mortar coupling. SIAM J. Numer. Anal. 51(1), 526–554 (2013)MATHMathSciNetCrossRefGoogle Scholar
  28. 28.
    Quarteroni, A., Valli, A.: Numerical Approximation of Partial Differential Equations, Springer Series in Computational Mathematics, vol. 23. Springer, Berlin (1994)Google Scholar
  29. 29.
    Rannacher, R., Westenberger, A., Wollner, W.: Adaptive finite element solution of eigenvalue problems: balancing of discretization and iteration error. J. Numer. Math. 18(4), 303–327 (2010)MATHMathSciNetCrossRefGoogle Scholar
  30. 30.
    Rektorys, K.: Variational Methods in Mathematics, Science and Engineering. D. Reidel Publishing Co., Dordrecht (1977). Translated from the Czech by Michael BaschGoogle Scholar
  31. 31.
    Saad, Y., Schultz, M.H.: GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Stat. Comput. 7(3), 856–869 (1986)MATHMathSciNetCrossRefGoogle Scholar
  32. 32.
    Silvester, P.: Symmetric quadrature formulae for simplexes. Math. Comput. 24, 95–100 (1970)MATHMathSciNetCrossRefGoogle Scholar
  33. 33.
    Strakoš, Z., Tichý, P.: Error estimation in preconditioned conjugate gradients. BIT 45(4), 789–817 (2005)MATHMathSciNetCrossRefGoogle Scholar
  34. 34.
    Verfürth, R.: A review of a posteriori error estimation and adaptive mesh–refinement techniques. Teubner-Wiley, Stuttgart (1996)MATHGoogle Scholar
  35. 35.
    Wheeler, M.F., Yotov, I.: A posteriori error estimates for the mortar mixed finite element method. SIAM J. Numer. Anal. 43(3), 1021–1042 (2005)MATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Vít Dolejší
    • 1
  • Ivana Šebestová
    • 1
  • Martin Vohralík
    • 2
  1. 1.Faculty of Mathematics and PhysicsCharles University in PraguePraha 8Czech Republic
  2. 2.INRIA Paris-RocquencourtLe ChesnayFrance

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