Skip to main content
SpringerLink
Log in

Robust a Posteriori Error Estimates for Conforming Discretizations of Diffusion Problems with Discontinuous Coefficients on Anisotropic Meshes

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

Abstract

In this paper, we study a posteriori estimates for different numerical methods of diffusion problems with discontinuous coefficients on anisotropic meshes, in particular, which can be applied to vertex-centered and cell-centered finite volume, finite difference and piecewise linear finite element methods. Based on the stretching ratios of the mesh elements, we improve a posteriori estimates developed by Vohralík (J Sci Comput 46:397–438, 2011), which are reliable and efficient on isotropic meshes but fail on anisotropic ones (see the numerical results of the paper). Without the assumption that the meshes are shape-regular, the resulting mesh-dependent error estimators are shown to be reliable and efficient with respect to 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, as long as the anisotropic mesh sufficiently reflects the anisotropy of the solution. In other words, they are equivalent to the estimates of Vohralík in the case of isotropic meshes and proved to be robust on anisotropic meshes as well. Based on \(\mathbf{H}(\mathrm {div})\)-conforming, locally conservative flux reconstruction, we suggest two different constructions of the equilibrated flux with the anisotropy of mesh, which is essential to the robustness of our estimates on anisotropic meshes. Numerical experiments in 2D confirm that our estimates are reliable and efficient on anisotropic meshes.

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.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14

Similar content being viewed by others

References

  1. Afif, M., Bergam, A., Mghazli, Z., Verfürth, R.: A posteriori estimators for the finite volume discretization of an elliptic equation. Numer. Algorithms 34, 127–136 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  2. Afif, M., Amaziane, B., Kunert, G., Mghazli, Z., Nicaise, S.: A posteriori error estimation for a finite volume discretization on anisotropic meshes. J. Sci. Comput. 43, 183–200 (2010)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  4. Ainsworth, M., Oden, J.T.: A Posteriori Error Estimation in Finite Element Analysis. Wiley, New York (2000)

    Book  MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  6. Apel, T., Nicaise, S., Sirch, D.: A posteriori error estimation of residual type for anisotropic diffusion–convection–reaction problems. J. Comput. Appl. Math. 235, 2805–2820 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  7. Arbogast, T., Chen, Z.: On the implementation of mixed methods as nonconforming methods for second-order elliptic problems. Math. Comput. 64, 943–972 (1995)

    MATH  MathSciNet  Google Scholar 

  8. Arnold, D.N., Brezzi, F.: Mixed and nonconforming finite element methods: implementation, postprocessing and error estimates. RAIRO Modél. Math. Anal. Numér. 19, 7–32 (1985)

    MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  10. Brezzi, F., Fortin, M.: Mixed and Hybrid Finite Element Methods. Springer, New York (1991)

    Book  MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  12. Chaillou, A.L., Suri, M.: Computable error estimators for the approximation of nonlinear problems by linearized models. Comput. Methods Appl. Mech. Eng. 196, 210–224 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  13. Chaillou, A.L., Suri, M.: A posteriori estimation of the linearization error for strongly monotone nonlinear operators. J. Comput. Appl. Math. 205, 72–87 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  14. 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 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  16. 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)

    Article  MATH  Google Scholar 

  17. 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. 200, 2782–2795 (2011)

    Article  MATH  Google Scholar 

  18. 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, 114–130 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  19. 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)

    Article  MATH  MathSciNet  Google Scholar 

  20. Grosman, S.: The Robustness of the Hierarchical a Posteriori Error Estimator for Reaction–Diffusion Equation on Anisotropic Meshes. SFB393-Preprint 2, Technische Universität Chemnitz, SFB 393 (Germany) (2004)

  21. Grosman, S.: An equilibrated residual method with a computable error approximation for a singularly perturbed reaction–diffusion problem on anisotropic finite element meshes. ESAIM Math. Model. Numer. Anal. 40, 239–267 (2006)

  22. Kunert, G.: A Posteriori Error Estimation for Anisotropic Tetrahedral and Triangular Finite Element Meshes. Logos Verlag, Berlin (1999). Also PhD thesis, TU Chemnitz. http://archiv.tu-chemnitz.de/pub/1999/0012/index.html

  23. Kunert, G.: An a posteriori residual error estimator for the finite element method on anisotropic tetrahedral meshes. Numer. Math. 86, 471–490 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  24. Kunert, G.: A local problem error estimator for anisotropic tetrahedral finite element meshes. SIAM J. Numer. Anal. 39, 668–689 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  25. Kunert, G., Nicaise, S.: Zienkiewicz–Zhu error estimators on anisotropic tetrahedral and triangular finite element meshes. ESAIM. Math. Model. Numer. Anal. 37, 1013–1043 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  26. Kunert, G., Verfürth, R.: Edge residuals dominate a posteriori error estimates for linear finite element methods on anisotropic triangular and tetrahedral meshes. Numer. Math. 86, 283–303 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  27. Mackenzie, J.A., Mayers, D.F., Mayfield, A.J.: Error estimates and mesh adaption for a cell vertex finite volume scheme. Notes Numer. Fluid Mech. 44, 290–310 (1993)

    Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  29. Picasso, M.: An anisotropic error indicator based on Zienkiewicz–Zhu error estimator: application to elliptic and parabolic problems. SIAM J. Sci. Comput. 24, 1328–1355 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  30. 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 

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

  32. Roberts, J.E., Thomas, J.M.: Mixed and Hybrid Methods. North-Holland, Amsterdam (1991)

    Book  Google Scholar 

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

    MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  35. Vohralík, M.: On the discrete Poincaré–Friedrichs inequalities for nonconforming approximations of the Sobolev space \(H^1\). Numer. Funct. Anal. Optim. 26, 925–952 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  36. Vohralík, M.: A posteriori error estimates for lowest-order mixed finite element discretizations of convection–diffusion–reactiion equations. SIAM J. Numer. Anal. 45, 1570–1599 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  37. 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)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgments

We would like to thank the anonymous referee for his helpful suggestions.

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Zhengzhou University, Zhengzhou, 450001, People’s Republic of China

    Jikun Zhao, Shaochun Chen & Bei Zhang

  2. LSEC and Institute of Computational Mathematics, Chinese Academy of Sciences, P.O. Box 2719, Beijing, 100190, People’s Republic of China

    Shipeng Mao

Authors
  1. Jikun Zhao
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Shaochun Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Bei Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Shipeng Mao
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jikun Zhao.

Additional information

This work is supported by National Natural Science Foundation of China (No. 11371331, 11101414, 11471329 and 91130026).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhao, J., Chen, S., Zhang, B. et al. Robust a Posteriori Error Estimates for Conforming Discretizations of Diffusion Problems with Discontinuous Coefficients on Anisotropic Meshes. J Sci Comput 64, 368–400 (2015). https://doi.org/10.1007/s10915-014-9937-7

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10915-014-9937-7

Keywords

Mathematics Subject Classification

Search

Navigation