Journal of Scientific Computing

, Volume 75, Issue 2, pp 1079–1101 | Cite as

Guaranteed A Posteriori Error Estimates for a Staggered Discontinuous Galerkin Method

Article
  • 196 Downloads

Abstract

In this paper, we present for the first time guaranteed upper bounds for the staggered discontinuous Galerkin method for diffusion problems. Two error estimators are proposed for arbitrary polynomial degrees and provide an upper bound on the energy error of the scalar unknown and \(L^2\)-error of the flux, respectively. Both error estimators are based on the potential and flux reconstructions. The potential reconstruction is given by a simple averaging operator. The equilibrated flux reconstruction can be found by solving local Neumann problems on elements sharing an edge with a Raviart–Thomas mixed method. Reliability and efficiency of the two a posteriori error estimators are proved. Numerical results are presented to validate the theoretical results.

Keywords

Staggered grid Discontinuous Galerkin method Guaranteed upper bound A posteriori error estimators 

References

  1. 1.
    Ainsworth, M., Oden, J.T.: A Posteriori Error Estimation in Finite Element Analysis. Wiley, New York (2000)CrossRefMATHGoogle Scholar
  2. 2.
    Alonso, A.: Error estimators for a mixed method. Numer. Math. 74, 385–395 (1996)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Arbogast, T., Chen, Z.: On the implementation of mixed methods as nonconforming methods for second-order elliptic problems. Math. Comput. 64, 943–972 (1995)MathSciNetMATHGoogle Scholar
  4. 4.
    Arnold, A.N., Brezzi, F.: Mixed and nonconforming finite element methods: imlementation, postprocessing and error estimates. RAIRO Modél. Math. Anal. Numér. Anal. 19, 7–32 (1985)CrossRefMATHGoogle Scholar
  5. 5.
    Babuška, I., Rheinboldt, W.C.: Error estimates for adaptive finite element computations. SIAM J. Numer. Anal. 15, 736–754 (1978)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Babuška, I., Rheinboldt, W.C.: A posteriori error estimates for the fintie element method. Int. J. Numer. Methods Eng. 12, 1597–1615 (1978)CrossRefMATHGoogle Scholar
  7. 7.
    Bernardi, C., Verfürth, R.: Adaptive finite element methods for elliptic equations with non-smooth coefficients. Numer. Math. 85, 579–608 (2000)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Braess, D., Schöberl, J.: Equilibrated residual error estimator for edge elements. Math. Comput. 77, 651–672 (2008)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Braess, D., Verfürth, R.: A posteriori error estimators for the Raviart–Thomas element. SIAM J. Numer. Anal. 33, 2431–2444 (1996)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Carstensen, C.: A posteriori error estimates for the mixed finite element method. Math. Comput. 66, 465–476 (1997)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Carstensen, C., Feischl, M., Page, M., Praetorius, D.: Axioms of adaptivity. Comput. Math. Appl. 67, 1195–1253 (2014)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Carstensen, C., Kim, D., Park, E.-J.: A priori and a posteriori pseudostress-velocity mixed finite element error analysis for the Stokes problem. SIAM J. Numer. Anal. 49, 2501–2523 (2011)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Carstensen, C., Park, E.-J.: Convergence and optimality of adaptive least squares finite element methods. SIAM J. Numer. Anal. 53(1), 43–62 (2015)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Chan, H.N., Chung, E.T., Cohen, G.: Stability and dispersion analysis of staggered discontinuous Galerkin method for wave propagation. Int. J. Numer. Model. 10, 233–256 (2013)MathSciNetMATHGoogle Scholar
  15. 15.
    Chen, Z., Dai, S.: On the efficiency of adaptive finite element methods for elliptic problems with discontinuous coefficients. SIAM J. Sci. Comput. 24, 443–462 (2002)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Chung, E.T., Ciarlet, P., Yu, T.F.: Convergence and superconvergence of staggered discontinuous Galerkin methods for the three-dimensional Maxwell’s equations on Cartesian grids. J. Comput. Phys. 235, 14–31 (2013)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Chung, E., Cockburn, B., Fu, G.: The staggered DG method is the limit of a hybridizable DG method. SIAM J. Numer. Anal. 52, 915–932 (2014)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Chung, E.T., Engquist, B.: Optimal discontinuous galerkin methods for wave propagation. SIAM J. Numer. Anal. 44, 2131–2158 (2006)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Chung, E.T., Engquist, B.: Optimal discontinuous galerkin methods for the acoustic wave equation in higher dimensions. SIAM J. Numer. Anal. 47, 3820–3848 (2009)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Chung, E.T., Kim, H.: A deluxe FETI-DP algorithm for a hybrid staggered discontinuous Galerkin method for H(curl)-elliptic problems. Int. J. Numer. Method Eng. 98, 1–23 (2014)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Chung, E.T., Kim, H.H., Widlund, O.: Two-level overlapping schwarz algorithms for a staggered discontinuous Galerkin method. SIAM J. Numer. Anal. 51, 47–67 (2013)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Chung, E.T., Lee, C.S.: A staggered discontinuous galerkin method for the curl–curl operator. IMA J. Numer. Anal. 32, 1241–1265 (2012)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Chung, E.T., Lee, C.S.: A staggered discontinuous galerkin method for the convection–diffusion equation. J. Numer. Math. 20, 1–31 (2012)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Chung, E.T., Yuen, M., Zhong, L.: A-posteriori error analysis for a staggered discontinuous Galerkin discretization of the time-harmonic Maxwell’s equations. Appl. Math. Comput. 237, 613–631 (2014)MathSciNetMATHGoogle Scholar
  25. 25.
    Dörfler, W., Wilderotter, O.: An adaptive finite element method for a linear elliptic equation with variable coefficients. Z. Angew. Math. Mech. 80, 481–491 (2000)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Ern, A., Nicaise, S., Vohralík, M.: An accurate H(div) flux reconstruction for discontinuous Galerkin approximations of elliptic problems. C.R. Math. Acad. Sci. Paris Ser. I(345), 709–712 (2007)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Ern, A., Vohralík, M.: Polynomial-degree-robust a posteriori estimates in a unified setting for conforming, nonconforming, discontinuous Galerkin, and mixed discretizations. SIAM J. Numer. Anal. 53, 1058–1081 (2015)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    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 Ser. I(347), 441–444 (2009)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Karakashian, O.A., Pascal, F.: A posteriori error estimates for a discontinuous Galerkin approximation of second-order elliptic problems. SIAM J. Numer. Anal. 41, 2374–2399 (2003)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Kim, D., Park, E.-J.: A posteriori error estimator for expanded mixed hybrid methods. Numer. Methods Partial Differ. Equ. 23, 973–988 (2007)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Kim, D., Park, E.-J.: A posteriori error estimators for the upstream weighting mixed methods for convection diffusion problems. Comput. Methods Appl. Mech. Eng. 197, 806–820 (2008)MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Kim, D., Park, E.-J.: A priori and a posteriori analysis of mixed finite element methods for nonlinear elliptic equations. SIAM J. Numer. Anal. 48, 1186–1207 (2010)MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Kim, H.H., Chung, E.T., Lee, C.S.: A staggered discontinuous Galerkin method for the Stokes system. SIAM J. Numer. Anal. 51, 3327–3350 (2013)MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Kim, H.H., Chung, E.T., Lee, C.S.: A BDDC algorithm for a class of staggered discontinuous Galerkin methods. Comput. Math. Appl. 67, 1373–1389 (2014)MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Kim, K.Y.: A posteriori error analysis for locally conservative mixed methods. Math. Comput. 76, 43–66 (2007)MathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    Kim, K.Y.: A posteriori error estimators for locally conservative methods of nonlinear elliptic problem. Appl. Numer. Math. 57, 1065–1080 (2007)MathSciNetCrossRefMATHGoogle Scholar
  37. 37.
    Ladevèze, P.: Comparaison de modèles de milieux continus. Ph.D. thesis, Université Pierre et Marie Curie, Paris6 (1975)Google Scholar
  38. 38.
    Larson, M.G., Målqvist, A.: A posteriori error estimates for mixed finite element approximations of elliptic problems. Numer. Math. 108, 487–500 (2008)MathSciNetCrossRefGoogle Scholar
  39. 39.
    Luce, E., Wohlmuth, B.I.: A local a posteriori error estimator based on equilibrated fluxes. SIAM J. Numer. Anal. 42, 1394–1414 (2004)MathSciNetCrossRefMATHGoogle Scholar
  40. 40.
    Prager, W., Synge, J.L.: Approximations in elasticity based on the concept of function space. Q. Appl. Math. 6, 241–269 (1947)MathSciNetCrossRefMATHGoogle Scholar
  41. 41.
    Rivère, B., Wheeler, M.F.: A posteriori error estimates for a discontinuous Galerkin method applied to elliptic problems. Comput. Math. Appl. 46, 141–163 (2003)MathSciNetCrossRefGoogle Scholar
  42. 42.
    Synge, J.L.: The Hypercircle in Mathematical Physics: A Method for the Approximate Solution of Boundary Value Problems. Cambridge University Press, New York (1957)MATHGoogle Scholar
  43. 43.
    Verfürth, R.: A Review of a Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques. Teubner-Wiley, Stuttgart (1996)MATHGoogle Scholar
  44. 44.
    Vohralík, M.: A posteriori error estimates for lowest-order mixed finite element discretization of convection–diffusion–reaction equations. SIAM J. Numer. Anal. 45, 1570–1599 (2007)MathSciNetCrossRefMATHGoogle Scholar
  45. 45.
    Vohralík, M.: Unified primal formulation-based a priori and a posteriori error analysis of mixed finite element methods. Math. Comput. 79, 2001–2032 (2010)MathSciNetCrossRefMATHGoogle Scholar
  46. 46.
    Vohralík, M.: Guaranteed and fully robust a posteriori error estimates for conforming discretizations of diffusion problems with discontinuous coefficients. J. Sci. Comput. 46, 397–438 (2011)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Department of MathematicsThe Chinese University of Hong KongShatinHong Kong, SAR
  2. 2.Department of Computational Science and EngineeringYonsei UniversitySeoulRepublic of Korea

Personalised recommendations