Skip to main content
SpringerLink
Log in

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

  • Published:
Journal of Scientific Computing Aims and scope Submit manuscript

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Ainsworth, M.: Robust a posteriori error estimation for nonconforming finite element approximation. SIAM J. Numer. Anal. 42(6), 2320–2341 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  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. 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. Angermann, L.: Balanced a posteriori error estimates for finite-volume type discretizations of convection-dominated elliptic problems. Computing 55(4), 305–323 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    MATH  MathSciNet  Google Scholar 

  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)

    MATH  MathSciNet  Google Scholar 

  7. Babuška, I.: Personal communication (2008)

  8. Babuška, I., Rheinboldt, W.C.: Error estimates for adaptive finite element computations. SIAM J. Numer. Anal. 15(4), 736–754 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  9. Bank, R.E., Rose, D.J.: Some error estimates for the box method. SIAM J. Numer. Anal. 24(4), 777–787 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  10. Bank, R.E., Weiser, A.: Some a posteriori error estimators for elliptic partial differential equations. Math. Comput. 44(170), 283–301 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  11. Bebendorf, M.: A note on the Poincaré inequality for convex domains. Z. Anal. Anwend. 22(4), 751–756 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  12. Bernardi, C., Verfürth, R.: Adaptive finite element methods for elliptic equations with non-smooth coefficients. Numer. Math. 85(4), 579–608 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  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

    Chapter  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  15. Braess, D., Schöberl, J.: Equilibrated residual error estimator for edge elements. Math. Comput. 77(262), 651–672 (2008)

    MATH  Google Scholar 

  16. Brezzi, F., Fortin, M.: Mixed and Hybrid Finite Element Methods. Springer Series in Computational Mathematics, vol. 15. Springer, New York (1991)

    MATH  Google Scholar 

  17. Cai, Z., Zhang, S.: Recovery-based error estimator for interface problems: conforming linear elements. SIAM J. Numer. Anal. 47(3), 2132–2156 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  24. Destuynder, P., Métivet, B.: Explicit error bounds in a conforming finite element method. Math. Comput. 68(228), 1379–1396 (1999)

    Article  MATH  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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. 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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    MATH  MathSciNet  Google Scholar 

  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. 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)

    Article  MATH  MathSciNet  Google Scholar 

  31. Fierro, F., Veeser, A.: A posteriori error estimators, gradient recovery by averaging, and superconvergence. Numer. Math. 103(2), 267–298 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  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)

  33. Haslinger, J., Hlaváček, I.: Convergence of a finite element method based on the dual variational formulation. Apl. Mat. 21(1), 43–65 (1976)

    MATH  MathSciNet  Google Scholar 

  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

    MATH  Google Scholar 

  35. Korotov, S.: Two-sided a posteriori error estimates for linear elliptic problems with mixed boundary conditions. Appl. Math. 52(3), 235–249 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  36. Ladevèze, P.: Comparaison de modèles de milieux continus. PhD thesis, Université Pierre et Marie Curie, Paris 6 (1975)

  37. Ladevèze, P., Leguillon, D.: Error estimate procedure in the finite element method and applications. SIAM J. Numer. Anal. 20(3), 485–509 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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. 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. Payne, L.E., Weinberger, H.F.: An optimal Poincaré inequality for convex domains. Arch. Ration. Mech. Anal. 5, 286–292 (1960)

    Article  MATH  MathSciNet  Google Scholar 

  43. Petzoldt, M.: A posteriori error estimators for elliptic equations with discontinuous coefficients. Adv. Comput. Math. 16(1), 47–75 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  44. Prager, W., Synge, J.L.: Approximations in elasticity based on the concept of function space. Q. Appl. Math. 5, 241–269 (1947)

    MATH  MathSciNet  Google Scholar 

  45. Putti, M., Cordes, C.: Finite element approximation of the diffusion operator on tetrahedra. SIAM J. Sci. Comput. 19(4), 1154–1168 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  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. Repin, S.I.: A posteriori Estimates for Partial Differential Equations. Radon Series on Computational and Applied Mathematics, vol. 4. de Gruyter, Berlin (2008)

    MATH  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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. Schöberl, J.: Personal communication (2009)

  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)

    MATH  Google Scholar 

  52. Vacek, J.: Dual variational principles for an elliptic partial differential equation. Apl. Mat. 21(1), 5–27 (1976)

    MATH  MathSciNet  Google Scholar 

  53. Vejchodský, T.: Guaranteed and locally computable a posteriori error estimate. IMA J. Numer. Anal. 26(3), 525–540 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  54. Verfürth, R.: A Review of a posteriori Error Estimation and Adaptive Mesh-Refinement Techniques. Teubner-Wiley, Stuttgart (1996)

    MATH  Google Scholar 

  55. Verfürth, R.: Robust a posteriori error estimates for stationary convection-diffusion equations. SIAM J. Numer. Anal. 43(4), 1766–1782 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    MATH  MathSciNet  Google Scholar 

  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. 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). www.hydroexpert.com

  61. Xu, J., Zhu, Y., Zou, Q.: New adaptive finite volume methods and convergence analysis. Preprint AM296, Mathematics Department, Penn State (2006)

  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)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Laboratoire Jacques-Louis Lions, UPMC Univ. Paris 06, UMR 7598, 75005, Paris, France

    Martin Vohralík

  2. Laboratoire Jacques-Louis Lions, CNRS, UMR 7598, 75005, Paris, France

    Martin Vohralík

Authors
  1. Martin Vohralík
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Martin Vohralík.

Additional information

This work was supported by the GNR MoMaS project “Numerical Simulations and Mathematical Modeling of Underground Nuclear Waste Disposal”, PACEN/CNRS, ANDRA, BRGM, CEA, EdF, IRSN, France.

Rights and permissions

Reprints and permissions

About this article

Cite this article

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). https://doi.org/10.1007/s10915-010-9410-1

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10915-010-9410-1

Keywords

Search

Navigation