Skip to main content

Equilibration a Posteriori Error Estimation for Convection–Diffusion–Reaction Problems

Abstract

We study a posteriori error estimates for convection–diffusion–reaction problems with possibly dominating convection or reaction and inhomogeneous boundary conditions. For the conforming FEM discretisation with streamline diffusion stabilisation, we derive reliable and efficient error estimators based on the reconstruction of equilibrated fluxes in an admissible discrete subspace of \({H({{\mathrm{div}}},\Omega )}\). Error estimators of this type have become popular recently since they provide guaranteed error bounds without further unknown constants. The estimators can be improved significantly by some postprocessing and divergence correction technique. For an extension of the energy norm by a dual norm of the convection part of the differential operator, robustness of the error estimator with respect to the coefficients of the problem is achieved. Numerical benchmarks illustrate the good performance of the error estimators for singularly perturbed problems, in particular with dominating convection.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8

References

  1. Ainsworth, M., Allendes, A., Barrenechea, G.R., Rankin, R.: Fully computable a posteriori error bounds for stabilised FEM approximations of convection–reaction–diffusion problems in three dimensions. Int. J. Numer. Methods Fluids 73(9), 765–790 (2013)

    MathSciNet  Google Scholar 

  2. Brenner, S.C., Carstensen, C.: Finite Element Methods. Wiley, London (2004)

    Book  Google Scholar 

  3. Bartels, S., Carstensen, C., Dolzmann, G.: Inhomogeneous Dirichlet conditions in a priori and a posteriori finite element error analysis. Numer. Math. 99(1), 1–24 (2004)

    MathSciNet  Article  MATH  Google Scholar 

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

    MathSciNet  Article  MATH  Google Scholar 

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

    MathSciNet  Article  MATH  Google Scholar 

  6. Carstensen, C., Eigel, M., Hoppe, R.H.W., Löbhard, C.: A review of unified a posteriori finite element error control. Numer. Math. Theory Methods Appl. 5(4), 509–558 (2012)

    MathSciNet  Article  MATH  Google Scholar 

  7. Cheddadi, I., Fučík, R., Prieto, M.I., Vohralík, M.: Guaranteed and robust a posteriori error estimates for singularly perturbed reaction–diffusion problems. Math. Model. Numer. Anal. 43(5), 867–888 (2009)

    MathSciNet  Article  MATH  Google Scholar 

  8. Ciarlet, P. G.: The Finite Element Method for Elliptic Problems. North-Holland Publishing Co., Amsterdam, 1978. Studies in Mathematics and its Applications, vol. 4

  9. Carstensen, C., Merdon, C.: Estimator competition for Poisson problems. J. Comput. Math. 28(3), 309–330 (2010). (electronic)

    MathSciNet  MATH  Google Scholar 

  10. Carstensen, C., Merdon, C.: Computational survey on a posteriori error estimators for nonconforming finite element methods for the Poisson problem. J. Comput. Appl. Math. 249, 74–94 (2013)

    MathSciNet  Article  MATH  Google Scholar 

  11. Carstensen, C., Merdon, C.: Effective postprocessing for equilibration a posteriori error estimators. Numer. Math. 123(3), 425–459 (2013)

    MathSciNet  Article  MATH  Google Scholar 

  12. Carstensen, C., Merdon, C.: Refined fully explicit a posteriori residual-based error control. SIAM J. Numer. Anal. 52(4), 1709–1728 (2014)

    MathSciNet  Article  MATH  Google Scholar 

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

    MathSciNet  Article  MATH  Google Scholar 

  14. Eriksson, K., Johnson, C.: Adaptive streamline diffusion finite element methods for stationary convection–diffusion problems. Math. Comput. 60(201), 167–188 (1993)

    MathSciNet  Article  MATH  Google Scholar 

  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)

    MathSciNet  Article  MATH  Google Scholar 

  16. Hughes, T.J.R., Brooks, A.: A multidimensional upwind scheme with no crosswind diffusion. In: Finite Element Methods for Convection Dominated Flows (Papers, Winter Ann. Meeting Am. Soc. Mech. Eng., New York, 1979), vol. 34 of AMD, pp. 19–35. Am. Soc. Mech. Eng., New York (1979)

  17. Hughes, T.J.R., Mallet, M., Akira, M.: A new finite element formulation for computational fluid dynamics. II. Beyond SUPG. Comput. Methods Appl. Mech. Eng. 54(3), 341–355 (1986)

    MathSciNet  Article  MATH  Google Scholar 

  18. John, V., Novo, J.: A robust SUPG norm a posteriori error estimator for stationary convection–diffusion equations. Math. Comput. Simul. 255(1), 289–305 (2013)

    MathSciNet  MATH  Google Scholar 

  19. Johnson, C., Nävert, U., Pitkäranta, J.: Finite element methods for linear hyperbolic problems. Comput. Methods Appl. Mech. Eng. 45(1–3), 285–312 (1984)

    MathSciNet  Article  MATH  Google Scholar 

  20. Johnson, C.: Adaptive finite element methods for diffusion and convection problems. Comput. Methods Appl. Mech. Eng. 82(1–3), 301–322 (1990)

    MathSciNet  Article  MATH  Google Scholar 

  21. John, Volker: A numerical study of a posteriori error estimators for convection–diffusion equations. Comput. Methods Appl. Mech. Eng. 190(5–7), 757–781 (2000)

    MathSciNet  Article  MATH  Google Scholar 

  22. Knabner, P., Angermann, L.: Numerical Methods for Elliptic and Parabolic Partial Differential Equations, Texts in Applied Mathematics, vol. 44. Springer, New York (2003)

    MATH  Google Scholar 

  23. Laugesen, R.S., Siudeja, B.A.: Minimizing Neumann fundamental tones of triangles: an optimal Poincaré inequality. J. Differ. Equ. 249(1), 118–135 (2010)

    MathSciNet  Article  MATH  Google Scholar 

  24. Luce, R., Wohlmuth, B.I.: A local a posteriori error estimator based on equilibrated fluxes. SIAM J. Numer. Anal. 42(4), 1394–1414 (2004). (electronic)

    MathSciNet  Article  MATH  Google Scholar 

  25. Merdon, C.: Aspects of guaranteed error control in computation for partial differential equations. Ph.D. thesis (2013)

  26. Papastavrou, A., Verfürth, R.: A posteriori error estimators for stationary convection–diffusion problems: a computational comparison. Comput. Methods Appl. Mech. Eng. 189(2), 449–462 (2000)

    MathSciNet  Article  MATH  Google Scholar 

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

    MathSciNet  Article  MATH  Google Scholar 

  28. Repin, S.: A Posteriori Estimates for Partial Differential Equations. Radon Series on Computational and Applied Mathematics, vol. 4. Walter de Gruyter GmbH & Co. KG, Berlin (2008)

    Book  Google Scholar 

  29. Repin, S., Sauter, S.: Functional a posteriori estimates for the reaction–diffusion problem. C. R. Math. Acad. Sci. Paris 343(5), 349–354 (2006)

    MathSciNet  Article  MATH  Google Scholar 

  30. Sangalli, G.: A uniform analysis of nonsymmetric and coercive linear operators. SIAM J. Math. Anal. 36(6), 2033–2048 (2005). (electronic)

    MathSciNet  Article  MATH  Google Scholar 

  31. Stevenson, R.: The uniform saturation property for a singularly perturbed reaction–diffusion equation. Numer. Math. 101, 355–379 (2005). (electronic)

    MathSciNet  Article  MATH  Google Scholar 

  32. Verfürth, R.: A posteriori error estimators for convection–diffusion equations. Numer. Math. 80(4), 641–663 (1998)

    MathSciNet  Article  MATH  Google Scholar 

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

    MathSciNet  Article  MATH  Google Scholar 

  34. Verfürth, R.: A note on constant-free a posteriori error estimates. SIAM J. Numer. Anal. 47(4), 3180–3194 (2009)

    MathSciNet  Article  MATH  Google Scholar 

  35. 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). (electronic)

    MathSciNet  Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to C. Merdon.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Eigel, M., Merdon, C. Equilibration a Posteriori Error Estimation for Convection–Diffusion–Reaction Problems. J Sci Comput 67, 747–768 (2016). https://doi.org/10.1007/s10915-015-0108-2

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10915-015-0108-2

Keywords

  • A posteriori
  • Error analysis
  • Finite element method
  • Equilibrated
  • Convection dominated
  • Adaptivity
  • Inhomogeneous Dirichlet
  • Augmented norm

Mathematics Subject Classification

  • 65N30
  • 65N15
  • 65J15
  • 65N22
  • 65J10