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A Numerical Study of Averaging Error Indicators in p-FEM

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Spectral and High Order Methods for Partial Differential Equations - ICOSAHOM 2012

Part of the book series: Lecture Notes in Computational Science and Engineering ((LNCSE,volume 95))

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Abstract

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).

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References

  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 http://dx.doi.org/10.1007/BF01385730

  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 http://dx.doi.org/10.1002/9781118032824

  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.1620280912

    Article  MathSciNet  MATH  Google Scholar 

  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 http://dx.doi.org/10.1007/PL00005387

  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 http://dx.doi.org/10.1137/1036141

  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.1620360402

    Article  MathSciNet  MATH  Google Scholar 

  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/S0025-5718-02-01412-6

    Google Scholar 

  8. Bernardi, C.: Indicateurs d’erreur en h-N version des éléments spectraux. RAIRO Modél. Math. Anal. Numér. 30(1), 1–38 (1996)

    MathSciNet  MATH  Google Scholar 

  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.2008.12.010

    Article  MathSciNet  MATH  Google Scholar 

  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 http://dx.doi.org/10.2307/2007465

  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/S003614299732334X

    Article  MathSciNet  MATH  Google Scholar 

  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:1014268310921

    Google Scholar 

  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. 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.1630030311

    Article  MATH  Google Scholar 

  15. Rodríguez, R.: Some remarks on Zienkiewicz-Zhu estimator. Numer. Methods Partial Differential Equations 10(5), 625–635 (1994). DOI 10.1002/num.1690100509

    Article  MathSciNet  Google Scholar 

  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.1630010103

    Article  MATH  Google Scholar 

  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.1620240206

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The first author gratefully acknowledges support by the ETH Foundation.

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Authors and Affiliations

  1. ETH Zürich, Rämistrasse 101, CH-8092, Zürich, Switzerland

    Philipp Dörsek

  2. Vienna University of Technology, Wiedner Hauptstraße 8–10, A-1040, Vienna, Austria

    J. Markus Melenk

Authors
  1. Philipp Dörsek
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  2. J. Markus Melenk
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Corresponding author

Correspondence to Philipp Dörsek .

Editor information

Editors and Affiliations

  1. Institut Polytechnique de Bordeaux, Université de Bordeaux, Pessac, France

    Mejdi Azaïez

  2. Ecole Nationale d'Ingénieurs de Tunis, Université de Tunis El Manar, Tunis, Tunisia

    Henda El Fekih

  3. EPFL-SB-MATHICSE-MCSS, Ecole Polytechnique Fédérale de Lausanne, Lausanne, Switzerland

    Jan S. Hesthaven

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Dörsek, P., Melenk, J.M. (2014). A Numerical Study of Averaging Error Indicators in p-FEM. In: Azaïez, M., El Fekih, H., Hesthaven, J. (eds) Spectral and High Order Methods for Partial Differential Equations - ICOSAHOM 2012. Lecture Notes in Computational Science and Engineering, vol 95. Springer, Cham. https://doi.org/10.1007/978-3-319-01601-6_18

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