Skip to main content
SpringerLink
Log in

A Posteriori Error Estimation by a Q 1 ∕ Q 2 Hierarchical Basis

  • Conference paper
  • First Online:
Numerical Mathematics and Advanced Applications 2011

  • 1991 Accesses

Abstract

This work presents an a posteriori error estimation technique for Q 1 finite elements on quadrilateral triangulations by residual evaluations with respect to biquadratic test functions. The localization is performed in terms of nodal error indicators instead of cell contributions. The reliability and efficiency of the estimator is shown. Further, we discuss a simplified estimator which is even more attractive from the computational point of view.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 169.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 219.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 219.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. M. Ainsworth and J. T. Oden. A unified approach to a posteriori error estimation using element residual methods. Numer. Math., 65(1):23–50, 1993.

    Article  MathSciNet  MATH  Google Scholar 

  2. M. Ainsworth and J. T. Oden. A Posteriori Error Estimation in Finite Element Analysis. Wiley Interscience, New York, 2000.

    Book  MATH  Google Scholar 

  3. I. Babuska and W. C. Rheinboldt. Error estimates for adaptive finite element computations. SIAM J. Numer. Anal., 15:736–754, 1978.

    Article  MathSciNet  MATH  Google Scholar 

  4. R.E. Bank. Hierarchical bases and the finite element method. Acta Numer., 44:1–43, 1996.

    Article  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  6. W. Dörfler. A convergent adaptive algorithm for Poisson’s equation. SIAM J. Numer. Anal., 33(3):1106–1124, 1996.

    Article  MathSciNet  MATH  Google Scholar 

  7. I. Babuška and T. Strouboulis. The Finite Element Method and its Reliability. Oxford Univers. Press, 2001.

    Google Scholar 

  8. R. Verfürth. A posteriori error estimates for nonlinear problems. finite element discretization of elliptic equations. Math. Comput., 62:445–475, 1994.

    Google Scholar 

  9. R. Verfürth. A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques. John Wiley/Teubner, New York-Stuttgart, 1996.

    MATH  Google Scholar 

  10. D. Yu. Asymptotically exact a posteriori error estimators for elements of bi-odd degree. Chinese J. Num. Math. and Appl., 13:64–78, 1991.

    Google Scholar 

Download references

Acknowledgements

We gratefully acknowledge the partial support of this work by the DFG Priority Program SPP 1276 (MetStröm).

Author information

Authors and Affiliations

  1. Mathematisches Seminar, Christian-Albrechts-Universität zu Kiel, Ludewig-Meyn-Str. 4, D-24098, Kiel, Germany

    M. Braack & N. Taschenberger

Authors
  1. M. Braack
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. N. Taschenberger
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to M. Braack .

Editor information

Editors and Affiliations

  1. , Department of Mathematics, University of Leicester, Leicester, LE!1 7RH, United Kingdom

    Andrea Cangiani

  2. , Department of Mathematics, University of Leicester, Leicester, LE1 7RL, United Kingdom

    Ruslan L. Davidchack

  3. , Department of Mathematics, University of Leicester, Leicester, LE1 7RH, United Kingdom

    Emmanuil Georgoulis

  4. , Department of Mathematics, University of Leicester, Leicester, LE1 7RH, United Kingdom

    Alexander N. Gorban

  5. , Department of Mathematics, University of Leicester, Leicester, LE1 7RH, United Kingdom

    Jeremy Levesley

  6. , School of Mathematical Sciences, University of Nottingham, University Park, Nottingham, NG7 2RD, United Kingdom

    Michael V. Tretyakov

Rights and permissions

Reprints and permissions

Copyright information

© 2013 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Braack, M., Taschenberger, N. (2013). A Posteriori Error Estimation by a Q 1 ∕ Q 2 Hierarchical Basis. In: Cangiani, A., Davidchack, R., Georgoulis, E., Gorban, A., Levesley, J., Tretyakov, M. (eds) Numerical Mathematics and Advanced Applications 2011. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-33134-3_3

Download citation

Publish with us

Policies and ethics

Search

Navigation