Journal of Scientific Computing

, Volume 46, Issue 3, pp 397–438 | Cite as

Guaranteed and Fully Robust a posteriori Error Estimates for Conforming Discretizations of Diffusion Problems with Discontinuous Coefficients

  • Martin Vohralík


We study in this paper a posteriori error estimates for H 1-conforming numerical approximations of diffusion problems with a diffusion coefficient piecewise constant on the mesh cells but arbitrarily discontinuous across the interfaces between the cells. Our estimates give a global upper bound on the error measured either as the energy norm of the difference between the exact and approximate solutions, or as a dual norm of the residual. They are guaranteed, meaning that they feature no undetermined constants. (Local) lower bounds for the error are also derived. Herein, only generic constants independent of the diffusion coefficient appear, whence our estimates are fully robust with respect to the jumps in the diffusion coefficient. In particular, no condition on the diffusion coefficient like its monotonous increasing along paths around mesh vertices is imposed, whence the present results also include the cases with singular solutions. For the energy error setting, the key requirement turns out to be that the diffusion coefficient is piecewise constant on dual cells associated with the vertices of an original simplicial mesh and that harmonic averaging is used in the scheme. This is the usual case, e.g., for the cell-centered finite volume method, included in our analysis as well as the vertex-centered finite volume, finite difference, and continuous piecewise affine finite element ones. For the dual norm setting, no such a requirement is necessary. Our estimates are based on H(div)-conforming flux reconstruction obtained thanks to the local conservativity of all the studied methods on the dual grids, which we recall in the paper; mutual relations between the different methods are also recalled. Numerical experiments are presented in confirmation of the guaranteed upper bound, full robustness, and excellent efficiency of the derived estimators.


Finite volume method Finite element method Finite difference method Discontinuous coefficients Harmonic averaging A posteriori error estimates Guaranteed upper bound Robustness 


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  1. 1.
    Ainsworth, M.: Robust a posteriori error estimation for nonconforming finite element approximation. SIAM J. Numer. Anal. 42(6), 2320–2341 (2005) MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Ainsworth, M.: A synthesis of a posteriori error estimation techniques for conforming, non-conforming and discontinuous Galerkin finite element methods. In: Recent Advances in Adaptive Computation. Contemp. Math., vol. 383, pp. 1–14. Am. Math. Soc., Providence (2005) Google Scholar
  3. 3.
    Ainsworth, M., Oden, J.T.: A posteriori error estimation in finite element analysis. In: Pure and Applied Mathematics. Wiley-Interscience/Wiley, New York (2000) Google Scholar
  4. 4.
    Angermann, L.: Balanced a posteriori error estimates for finite-volume type discretizations of convection-dominated elliptic problems. Computing 55(4), 305–323 (1995) MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Arbogast, T., Chen, Z.: On the implementation of mixed methods as nonconforming methods for second-order elliptic problems. Math. Comput. 64(211), 943–972 (1995) MATHMathSciNetGoogle Scholar
  6. 6.
    Arnold, D.N., Brezzi, F.: Mixed and nonconforming finite element methods: implementation, postprocessing and error estimates. RAIRO Modél. Math. Anal. Numér. 19(1), 7–32 (1985) MATHMathSciNetGoogle Scholar
  7. 7.
    Babuška, I.: Personal communication (2008) Google Scholar
  8. 8.
    Babuška, I., Rheinboldt, W.C.: Error estimates for adaptive finite element computations. SIAM J. Numer. Anal. 15(4), 736–754 (1978) MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Bank, R.E., Rose, D.J.: Some error estimates for the box method. SIAM J. Numer. Anal. 24(4), 777–787 (1987) MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Bank, R.E., Weiser, A.: Some a posteriori error estimators for elliptic partial differential equations. Math. Comput. 44(170), 283–301 (1985) MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Bebendorf, M.: A note on the Poincaré inequality for convex domains. Z. Anal. Anwend. 22(4), 751–756 (2003) MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Bernardi, C., Verfürth, R.: Adaptive finite element methods for elliptic equations with non-smooth coefficients. Numer. Math. 85(4), 579–608 (2000) MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Braess, D.: Theory, fast solvers, and applications in elasticity theory. In: Finite Elements, 3rd edn. Cambridge University Press, Cambridge (2007). Translated from the German by Larry L. Schumaker CrossRefGoogle Scholar
  14. 14.
    Braess, D., Pillwein, V., Schöberl, J.: Equilibrated residual error estimates are p-robust. Comput. Methods Appl. Mech. Eng. 198(13–14), 1189–1197 (2009) MATHCrossRefGoogle Scholar
  15. 15.
    Braess, D., Schöberl, J.: Equilibrated residual error estimator for edge elements. Math. Comput. 77(262), 651–672 (2008) MATHGoogle Scholar
  16. 16.
    Brezzi, F., Fortin, M.: Mixed and Hybrid Finite Element Methods. Springer Series in Computational Mathematics, vol. 15. Springer, New York (1991) MATHGoogle Scholar
  17. 17.
    Cai, Z., Zhang, S.: Recovery-based error estimator for interface problems: conforming linear elements. SIAM J. Numer. Anal. 47(3), 2132–2156 (2009) MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Carstensen, C., Funken, S.A.: Constants in Clément-interpolation error and residual based a posteriori error estimates in finite element methods. East-West J. Numer. Math. 8(3), 153–175 (2000) MATHMathSciNetGoogle Scholar
  19. 19.
    Chaillou, A.L., Suri, M.: Computable error estimators for the approximation of nonlinear problems by linearized models. Comput. Methods Appl. Mech. Eng. 196(1–3), 210–224 (2006) MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Chaillou, A.L., Suri, M.: A posteriori estimation of the linearization error for strongly monotone nonlinear operators. J. Comput. Appl. Math. 205(1), 72–87 (2007) MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    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
  22. 22.
    Cheddadi, I., Fučík, R., Prieto, M.I., Vohralík, M.: Guaranteed and robust a posteriori error estimates for singularly perturbed reaction–diffusion problems. Modél. Math. Anal. Numér. 43(5), 867–888 (2009) MATHCrossRefGoogle Scholar
  23. 23.
    Chen, Z., Dai, S.: On the efficiency of adaptive finite element methods for elliptic problems with discontinuous coefficients. SIAM J. Sci. Comput. 24(2), 443–462 (2002) MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Destuynder, P., Métivet, B.: Explicit error bounds in a conforming finite element method. Math. Comput. 68(228), 1379–1396 (1999) MATHCrossRefGoogle 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(7), 481–491 (2000) MATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    El Alaoui, L., Ern, A., Vohralík, M.: Guaranteed and robust a posteriori error estimates and balancing discretization and linearization errors for monotone nonlinear problems. Comput. Methods Appl. Mech. Eng. (2010). doi: 10.1016/j.cma.2010.03.024 Google Scholar
  27. 27.
    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) MATHCrossRefMathSciNetGoogle 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 347, 441–444 (2009) MATHMathSciNetGoogle Scholar
  29. 29.
    Eymard, R., Gallouët, T., Herbin, R.: Finite volume methods. In: Handbook of Numerical Analysis, vol. VII, pp. 713–1020. North-Holland, Amsterdam (2000) Google Scholar
  30. 30.
    Eymard, R., Hilhorst, D., Vohralík, M.: A combined finite volume–nonconforming/mixed-hybrid finite element scheme for degenerate parabolic problems. Numer. Math. 105(1), 73–131 (2006) MATHCrossRefMathSciNetGoogle Scholar
  31. 31.
    Fierro, F., Veeser, A.: A posteriori error estimators, gradient recovery by averaging, and superconvergence. Numer. Math. 103(2), 267–298 (2006) MATHCrossRefMathSciNetGoogle Scholar
  32. 32.
    Hannukainen, A., Stenberg, R., Vohralík, M.: A unified framework for a posteriori error estimation for the Stokes problem. Preprint R10016, Laboratoire Jacques-Louis Lions & HAL Preprint 00470131, submitted for publication (2010) Google Scholar
  33. 33.
    Haslinger, J., Hlaváček, I.: Convergence of a finite element method based on the dual variational formulation. Apl. Mat. 21(1), 43–65 (1976) MATHMathSciNetGoogle Scholar
  34. 34.
    Hlaváček, I., Haslinger, J., Nečas, J., Lovíšek, J.: Solution of Variational Inequalities in Mechanics. Applied Mathematical Sciences, vol. 66. Springer, New York (1988). Translated from the Slovak by J. Jarník MATHGoogle Scholar
  35. 35.
    Korotov, S.: Two-sided a posteriori error estimates for linear elliptic problems with mixed boundary conditions. Appl. Math. 52(3), 235–249 (2007) MATHCrossRefMathSciNetGoogle Scholar
  36. 36.
    Ladevèze, P.: Comparaison de modèles de milieux continus. PhD thesis, Université Pierre et Marie Curie, Paris 6 (1975) Google Scholar
  37. 37.
    Ladevèze, P., Leguillon, D.: Error estimate procedure in the finite element method and applications. SIAM J. Numer. Anal. 20(3), 485–509 (1983) MATHCrossRefMathSciNetGoogle Scholar
  38. 38.
    Letniowski, F.W.: Three-dimensional Delaunay triangulations for finite element approximations to a second-order diffusion operator. SIAM J. Sci. Stat. Comput. 13(3), 765–770 (1992) MATHCrossRefMathSciNetGoogle Scholar
  39. 39.
    Luce, R., Wohlmuth, B.I.: A local a posteriori error estimator based on equilibrated fluxes. SIAM J. Numer. Anal. 42(4), 1394–1414 (2004) MATHCrossRefMathSciNetGoogle Scholar
  40. 40.
    Nečas, J., Hlaváček, I.: Mathematical Theory of Elastic and Elasto-Plastic Bodies: an Introduction. Studies in Applied Mechanics, vol. 3. Elsevier Scientific, Amsterdam (1980) Google Scholar
  41. 41.
    Neittaanmäki, P., Repin, S.: Error control and a posteriori estimates. In: Reliable methods for computer simulation. Studies in Mathematics and its Applications, vol. 33. Elsevier Science B.V., Amsterdam (2004) Google Scholar
  42. 42.
    Payne, L.E., Weinberger, H.F.: An optimal Poincaré inequality for convex domains. Arch. Ration. Mech. Anal. 5, 286–292 (1960) MATHCrossRefMathSciNetGoogle Scholar
  43. 43.
    Petzoldt, M.: A posteriori error estimators for elliptic equations with discontinuous coefficients. Adv. Comput. Math. 16(1), 47–75 (2002) MATHCrossRefMathSciNetGoogle Scholar
  44. 44.
    Prager, W., Synge, J.L.: Approximations in elasticity based on the concept of function space. Q. Appl. Math. 5, 241–269 (1947) MATHMathSciNetGoogle Scholar
  45. 45.
    Putti, M., Cordes, C.: Finite element approximation of the diffusion operator on tetrahedra. SIAM J. Sci. Comput. 19(4), 1154–1168 (1998) MATHCrossRefMathSciNetGoogle Scholar
  46. 46.
    Repin, S.I.: A posteriori error estimation for nonlinear variational problems by duality theory. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. 28, 201–214 (1997); Kraev. Zadachi Mat. Fiz. i Smezh. Vopr. Teor. Funktsii. 342 Google Scholar
  47. 47.
    Repin, S.I.: A posteriori Estimates for Partial Differential Equations. Radon Series on Computational and Applied Mathematics, vol. 4. de Gruyter, Berlin (2008) MATHGoogle Scholar
  48. 48.
    Rivière, B., Wheeler, M.F., Banas, K.: Part II. Discontinuous Galerkin method applied to single phase flow in porous media. Comput. Geosci. 4(4), 337–349 (2000) MATHCrossRefGoogle Scholar
  49. 49.
    Roberts, J.E., Thomas, J.-M.: Mixed and hybrid methods. In: Handbook of Numerical Analysis, vol. II, pp. 523–639. North-Holland, Amsterdam (1991) Google Scholar
  50. 50.
    Schöberl, J.: Personal communication (2009) Google Scholar
  51. 51.
    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
  52. 52.
    Vacek, J.: Dual variational principles for an elliptic partial differential equation. Apl. Mat. 21(1), 5–27 (1976) MATHMathSciNetGoogle Scholar
  53. 53.
    Vejchodský, T.: Guaranteed and locally computable a posteriori error estimate. IMA J. Numer. Anal. 26(3), 525–540 (2006) MATHCrossRefMathSciNetGoogle Scholar
  54. 54.
    Verfürth, R.: A Review of a posteriori Error Estimation and Adaptive Mesh-Refinement Techniques. Teubner-Wiley, Stuttgart (1996) MATHGoogle Scholar
  55. 55.
    Verfürth, R.: Robust a posteriori error estimates for stationary convection-diffusion equations. SIAM J. Numer. Anal. 43(4), 1766–1782 (2005) MATHCrossRefMathSciNetGoogle Scholar
  56. 56.
    Vohralík, M.: On the discrete Poincaré–Friedrichs inequalities for nonconforming approximations of the Sobolev space H 1. Numer. Funct. Anal. Optim. 26(7–8), 925–952 (2005) MATHCrossRefMathSciNetGoogle Scholar
  57. 57.
    Vohralík, M.: A posteriori error estimates for lowest-order mixed finite element discretizations of convection-diffusion-reaction equations. SIAM J. Numer. Anal. 45(4), 1570–1599 (2007) MATHCrossRefMathSciNetGoogle Scholar
  58. 58.
    Vohralík, M.: A posteriori error estimation in the conforming finite element method based on its local conservativity and using local minimization. C. R. Math. Acad. Sci. Paris 346(11–12), 687–690 (2008) MATHMathSciNetGoogle Scholar
  59. 59.
    Vohralík, M.: Two types of guaranteed (and robust) a posteriori estimates for finite volume methods. In: Finite Volumes for Complex Applications V, pp. 649–656. ISTE/Wiley, London/Hoboken (2008) Google Scholar
  60. 60.
    Vohralík, M., Ramarosy, N.: Talisman, a finite volume–finite element tool for numerical simulation of subsurface flow and contaminant transport with a posteriori error control and adaptive mesh refinement. Presentation and User guide. Tech. rep., HydroExpert, 53 rue Charles Frérot, 94 250 Gentilly, France (2007).
  61. 61.
    Xu, J., Zhu, Y., Zou, Q.: New adaptive finite volume methods and convergence analysis. Preprint AM296, Mathematics Department, Penn State (2006) Google Scholar
  62. 62.
    Zienkiewicz, O.C., Zhu, J.Z.: A simple error estimator and adaptive procedure for practical engineering analysis. Int. J. Numer. Methods Eng. 24(2), 337–357 (1987) MATHCrossRefMathSciNetGoogle Scholar

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© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Laboratoire Jacques-Louis LionsUPMC Univ. Paris 06, UMR 7598ParisFrance
  2. 2.Laboratoire Jacques-Louis LionsCNRS, UMR 7598ParisFrance

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