Skip to main content
SpringerLink
Log in

A unifying theory of a posteriori finite element error control

  • Published:
Numerische Mathematik Aims and scope Submit manuscript

Summary

Residual-based a posteriori error estimates are derived within a unified setting for lowest-order conforming, nonconforming, and mixed finite element schemes. The various residuals are identified for all techniques and problems as the operator norm ||ℓ|| of a linear functional of the form

in the variable υ of a Sobolev space V. The main assumption is that the first-order finite element space is included in the kernel Ker ℓ of ℓ. As a consequence, any residual estimator that is a computable bound of ||ℓ|| can be used within the proposed frame without further analysis for nonconforming or mixed FE schemes. Applications are given for the Laplace, Stokes, and Navier-Lamè equations.

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., Oden, J.T.: A posteriori error estimation in finite element analysis. Wiley-Interscience [John Wiley & Sons], New York, 2000

  2. Alonso, A.: Error estimators for a mixed method. Numer. Math. 74(4), 385–395 (1996)

    Article  Google Scholar 

  3. Arnold, D.N., Falk, R.S.: A uniformly accurate finite element method for the Reissner-Mindlin plate model. SIAM J. Numer. Anal. 26, 1276–1290 (1989)

    Article  Google Scholar 

  4. Arnold, D.N., Brezzi, F., Douglas, J.: PEERS: A new finite element for plane elasticity. Japan J. Appl. Math 1(2), 347–367 (1984)

    Google Scholar 

  5. Babuška, I., Miller, A.: A feedback finite element method with a posteriori error estimation: Part I, The finite element method and some properties of the a posteriori estimator. Comp. Methods Appl. Mech. Engrg. 61(1), 1–40 (1987)

    Article  Google Scholar 

  6. Babuška, I., Rheinboldt, W.C.: A posteriori error analysis of finite element solutions for one–dimensional problems SIAM J. Numer. Anal. 18, 565–589 (1981)

    Article  Google Scholar 

  7. Babuška, I., Strouboulis, T.: The Finite Element Method and its Reliability. The Clarendon Press Oxford University Press, xii+802, 2001

  8. Bartels, S., Carstensen, C.: Each averaging technique yields reliable a posteriori error control in FEM on unstructured grids. II. Higher order FEM. Math. Comp. 71(239), 971–994 (2002)

    Google Scholar 

  9. Becker, R., Rannacher, R.: A feed-back approach to error control in finite element methods: basic analysis and examples. East-West J. Numer. Math 4, 237–264 (1996)

    Google Scholar 

  10. Becker, R., Rannacher, R.: An optimal control approach to a posteriori error estimation in finite element methods. Acta Numerica Cambridge University Press, 2001, 1–102

  11. Braess, D.: Finite Elements. Cambridge University Press, 1997

  12. Braess, D., Carstensen, C.,Reddy, B.D.: Uniform Convergence and a posteriori error estimators for the enhanced strain finite element method. Numer. Math. 96, 461–479 (2004)

    Article  Google Scholar 

  13. Brenner, S.C., Scott, L.R.: The mathematical theory of finite element methods. Springer Verlag, New York, Texts in Applied Mathematics 15, 1994

  14. Brezzi, F., Fortin, M.: Mixed and Hybrid Finite Element Methods. Springer-Verlag, New York, 1991

  15. Carstensen, C.: A posteriori error estimate for the mixed finite element method. Math. Comp. 66, 465–476 (1997)

    Article  Google Scholar 

  16. Carstensen, C., Bartels, S., Dolzmann, G.: A posteriori error estimates for nonconforming finite element methods. Numer. Math. 92(2), 233–256 (2002)

    Article  Google Scholar 

  17. Carstensen, C., Bartels, S., Klose, R.: An experimental survey of a posteriori Courant finite element error control for the Poisson equation. Adv. Comput. Math. 15(1–4), 79–106 (2002)

    Google Scholar 

  18. Carstensen, C., Dolzmann, G.: A posteriori error estimates for mixed FEM in elasticity. Numer. Math. 81(2) 187–209 (1998)

    Google Scholar 

  19. Carstensen, C., Dolzmann, G., Funken, S.A., Helm, D.S.: Locking-free adaptive mixed finite element methods in linear elasticity. Comput. Methods Appl. Mech. Engrg. 190(13–14), 1701–1718 (2000)

    Google Scholar 

  20. Carstensen, C., Funken, S.A.: A posteriori error control in low-order finite element discretisations of incompressible stationary flow problems. Math. Comp. 70(236), 1353–1381 (2001)

    Article  Google Scholar 

  21. Carstensen C., Funken S.A.: Averaging technique for FE-a posteriori error control in elasticity. Part I: Conforming FEM. Comput. Methods Appl. Mech. Engrg. 190, 2483–2498 (2001). Part II: λ-independent estimates. Comput. Methods Appl. Mech. Engrg. 190, 4663–4675 (2001). Part III: Locking-free conforming FEM. Comput. Methods Appl. Mech. Engrg. 191(8–10) 861–877 (2001)

    Google Scholar 

  22. Carstensen, C., Verfürth, R.: Edge residuals dominate a posteriori error estimates for low order finite element methods. SIAM J. Numer. Anal. 36(5), 1571–1587 (1999)

    Article  Google Scholar 

  23. Dari, E., Duran, R., Padra, C., Vampa, V.: A posteriori error estimators for nonconforming finite element methods. RAIRO Modél. Math. Anal. Numér. 30(4), 385–400 (1996)

    Google Scholar 

  24. Dari, E., Durdfn, R., Padra, C.: Error estimators for nonconforming finite element approximations of the Stokes problem. Math. Comp. 64(211), 1017–1033 (1995)

    Google Scholar 

  25. Eriksson, K., Johnson, C.: Adaptive finite element methods for parabolic problems I. A linear model problem. SIAM J. Numer. Anal. 28, 43–77 (1991)

    Article  Google Scholar 

  26. Hoppe, R.H.W., Wohlmuth, B.: Element-orientated and edge-orientated local error estimates for nonconforming finite element methods. Math. Modeling Numer. Anal. 30, 237–263 (1996)

    Google Scholar 

  27. Kouhia, R. and Stenberg, R.: A linear nonconforming finite element method for nearly incompressible elasticity and Stokes flow. Comput. Methods Appl. Mech. Engrg. 124(3), 195–212 (1995)

    Article  Google Scholar 

  28. Stenberg, R.: A family of mixed finite elements for the elasticity problem. Numer. Math. 53(5) 513–538 (1988)

    Google Scholar 

  29. Verfürth, R.: A posteriori error estimators for the Stokes equations. Numer. Math. 55(3), 309–325 (1989)

    Article  Google Scholar 

  30. Verfürth, R.: A review of a posteriori error estimation and adaptive mesh-refinement techniques, Wiley-Teubner (1996)

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Humboldt-Universität zu Berlin, Unter den Linden 6, 10099, Berlin, Germany

    C. Carstensen

Authors
  1. C. Carstensen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to C. Carstensen.

Additional information

Supported by the DFG Research Center Matheon “Mathematics for key technologies” in Berlin.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Carstensen, C. A unifying theory of a posteriori finite element error control. Numer. Math. 100, 617–637 (2005). https://doi.org/10.1007/s00211-004-0577-y

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00211-004-0577-y

Mathematics Subject Classification (2000)

Search

Navigation