A Numerical Study of Averaging Error Indicators in p-FEM

  • Philipp Dörsek
  • J. Markus Melenk
Conference paper
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 95)


We consider the averaging error indicator in the context of the p-FEM. We explain how a proof of reliability and efficiency might look, and why the error indicator will behave differently than for low order methods. Using two model problems, one with nonsmooth, the other one with smooth solution, we identify appropriate spaces for the averaged fluxes in order to obtain reasonable reliability and efficiency bounds on the averaging error indicator for p-FEM. In particular, averaging over two neighbouring elements using global polynomials of the same polynomial degree as the finite element solution leads to reliability and efficiency up to a factor of order O(p).



The first author gratefully acknowledges support by the ETH Foundation.


  1. 1.
    Ainsworth, M., Craig, A.: A posteriori error estimators in the finite element method. Numer. Math. 60(4), 429–463 (1992). DOI 10.1007/BF01385730. URL
  2. 2.
    Ainsworth, M., Oden, J.T.: A posteriori error estimation in finite element analysis. Pure and Applied Mathematics (New York). Wiley-Interscience [John Wiley & Sons], New York (2000). DOI 10.1002/9781118032824. URL
  3. 3.
    Ainsworth, M., Zhu, J.Z., Craig, A.W., Zienkiewicz, O.C.: Analysis of the Zienkiewicz-Zhu a posteriori error estimator in the finite element method. Internat. J. Numer. Methods Engrg. 28(9), 2161–2174 (1989). DOI 10.1002/nme.1620280912MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Babuška, I., Guo, B.: Optimal estimates for lower and upper bounds of approximation errors in the p-version of the finite element method in two dimensions. Numer. Math. 85(2), 219–255 (2000). DOI 10.1007/PL00005387. URL
  5. 5.
    Babuška, I., Suri, M.: The p and h-p versions of the finite element method, basic principles and properties. SIAM Rev. 36(4), 578–632 (1994). DOI 10.1137/1036141. URL
  6. 6.
    Babuška, I.M., Rodríguez, R.: The problem of the selection of an a posteriori error indicator based on smoothening techniques. Internat. J. Numer. Methods Engrg. 36(4), 539–567 (1993). DOI 10.1002/nme.1620360402MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    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 (electronic) (2002). DOI 10.1090/S0025-5718-02-01412-6Google Scholar
  8. 8.
    Bernardi, C.: Indicateurs d’erreur en h-N version des éléments spectraux. RAIRO Modél. Math. Anal. Numér. 30(1), 1–38 (1996)MathSciNetMATHGoogle Scholar
  9. 9.
    Braess, D., Pillwein, V., Schöberl, J.: Equilibrated residual error estimates are p-robust. Comput. Methods Appl. Mech. Engrg. 198(13–14), 1189–1197 (2009). DOI 10.1016/j.cma.2008.12.010MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Canuto, C., Quarteroni, A.: Approximation results for orthogonal polynomials in Sobolev spaces. Math. Comp. 38(157), 67–86 (1982). DOI 10.2307/2007465. URL
  11. 11.
    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). DOI 10.1137/S003614299732334XMathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Melenk, J.M., Wohlmuth, B.I.: On residual-based a posteriori error estimation in hp-FEM. Adv. Comput. Math. 15(1–4), 311–331 (2002) (2001). DOI 10.1023/A:1014268310921Google Scholar
  13. 13.
    Melenk, J.M., Wurzer, T.: On the stability of the polynomial L 2-projection on triangles and tetrahedra. Tech. rep., Institute for Analysis and Scientific Computing, Vienna University of Technology (2012)Google Scholar
  14. 14.
    Rank, E., Zienkiewicz, O.: A simple error estimator in the finite element method. Commun. Appl. Numer. Methods 3, 243–249 (1987). DOI 10.1002/cnm.1630030311CrossRefMATHGoogle Scholar
  15. 15.
    Rodríguez, R.: Some remarks on Zienkiewicz-Zhu estimator. Numer. Methods Partial Differential Equations 10(5), 625–635 (1994). DOI 10.1002/num.1690100509MathSciNetCrossRefGoogle Scholar
  16. 16.
    Zienkiewicz, O., Li, X.K., Nakazawa, S.: Iterative solution of mixed problems and the stress recovery procedures. Commun. Appl. Numer. Methods 1, 3–9 (1985). DOI 10.1002/cnm.1630010103CrossRefMATHGoogle Scholar
  17. 17.
    Zienkiewicz, O.C., Zhu, J.Z.: A simple error estimator and adaptive procedure for practical engineering analysis. Internat. J. Numer. Methods Engrg. 24(2), 337–357 (1987). DOI 10.1002/nme.1620240206MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.ETH ZürichZürichSwitzerland
  2. 2.Vienna University of TechnologyViennaAustria

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