Skip to main content
SpringerLink
Log in

Superconvergence of Some Linear and Quadratic Functionals for Higher-Order Finite Elements

  • Conference paper
  • First Online:
Finite Difference Methods,Theory and Applications (FDM 2014)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9045))

Included in the following conference series:

  • 2601 Accesses

Abstract

This paper deals with the calculation of linear and quadratic functionals of approximate solutions obtained by the finite element method. It is shown that under certain conditions the output functionals of an approximate solution are computed with higher order of accuracy than that of the solution itself. These abstract results are illustrated by two numerical examples for the Poisson equation.

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 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight 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. Shaydurov, V., Liu, T., Zheng, Z.: Four-stage computational technology with adaptive numerical methods for computational aerodynamics. Am. Inst. Phys. Conf. Proc. 1484, 42–48 (2012)

    Google Scholar 

  2. ADIGMA. A European Initiative on the Development of Adaptive Higher-Order Variational Methods for Aerospace Applications. Notes on Numerical Fluid Mechanics and Multidisciplinary Design. 113, Berlin: Springer (2010)

    Google Scholar 

  3. Bangerth, W., Rannacher, R.: Adaptive Finite Element Methods for Differential Equations. Birkhäuser, Berlin (2003)

    Book  MATH  Google Scholar 

  4. Yue, H., Shaydurov, V.: Superconvergence of some output functionals for Hermitian finite elements. Young Scientist 12, 15–19 (2012)

    Google Scholar 

  5. Strang, G., Fix, G.J.: An Analysis of the Finite Element Method. Prentice-Hall, New York (1973)

    MATH  Google Scholar 

  6. Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. North Holland, New York (1978)

    MATH  Google Scholar 

  7. Aubin, J.P.: Behavior of the error of the approximate solutions of boundary value problems for linear elliptic operators by Galerkin’s and finite difference methods. Ann. Scuola Norm. Sp. Pisa. 21, 599–637 (1967)

    MATH  MathSciNet  Google Scholar 

  8. Nitsche, J.A.: Ein Kriterium für die Quasi-Optimalitat des Ritzchen Verfahrens. Numer. Math. 11, 346–348 (1968)

    Article  MATH  MathSciNet  Google Scholar 

  9. Oganesyan, L.A.: Investigation of the convergence rate of variational-difference schemes for elliptic second order equations in a two-dimensional domain with a smooth boundary. Zh. Vychisl. Mat. i Mat. Fiz. 9, 1102–1120 (1969)

    MATH  Google Scholar 

  10. Adams, R.A., Fournier, J.J.F.: Sobolev spaces. Academic Press, New York (2003)

    MATH  Google Scholar 

  11. Chen, Z., Wu, H.: Selected Topics in Finite Element Methods. Science Press, Beijing (2010)

    Google Scholar 

  12. Shaydurov, V., Shut, S., Gileva, L.: Some properties of Hermite finite elements on rectangles. Am. Inst. Phys. Cof. Proc. 1629, 32–43 (2014)

    Google Scholar 

  13. Bogner, F.K., Fox, R.L., Schmit, L.A.: The generation of interelement compatible stiffness and mass matrices by the use of interpolation formulas. In: Proceedings of the Conference on Matrix Methods in Structural Mechanics, pp. 397–444. Wright-Patterson Air Force Base, Ohio (1965)

    Google Scholar 

  14. Zhang, S.: On the full C\(_{1}\)-Q\(_{k}\) finite element spaces on rectangles and cuboids. Adv. Appl. Math. Mech. 2(6), 701–721 (2010). doi:10.4208/aamm.09-m0993

    MATH  MathSciNet  Google Scholar 

  15. Kondrat’ev, V.A.: Boundary-value problems for elliptic equations in domain with conic and angular points. Trans. Moscow Math. Soc. 16, 209–292 (1967)

    MATH  Google Scholar 

  16. Gileva, L., Shaydurov, V., Dobronets, B.: The triangular Hermite finite element complementing the Bogner-Fox-Schmit rectangle. Appl. Math. 5(12A), 50–56 (2013)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Computational Modeling of Siberian Branch of Russian Academy of Sciences, Krasnoyarsk, Russia

    Vladimir Shaydurov

  2. Beihang University, Beijing, China

    Vladimir Shaydurov & Tianshi Xu

Authors
  1. Vladimir Shaydurov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Tianshi Xu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Vladimir Shaydurov .

Editor information

Editors and Affiliations

  1. Bulgarian Academy of Sciences, Institute of Information and Communication Technologies, Sofia, Bulgaria

    Ivan Dimov

  2. Institue of Mathematics and MTA-ELTE Numerical Analysis and Large Networks Research Group, Eötvös Loránd University, Budapest, Hungary

    István Faragó

  3. University of Rousse, Rousse, Bulgaria

    Lubin Vulkov

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer International Publishing Switzerland

About this paper

Cite this paper

Shaydurov, V., Xu, T. (2015). Superconvergence of Some Linear and Quadratic Functionals for Higher-Order Finite Elements. In: Dimov, I., Faragó, I., Vulkov, L. (eds) Finite Difference Methods,Theory and Applications. FDM 2014. Lecture Notes in Computer Science(), vol 9045. Springer, Cham. https://doi.org/10.1007/978-3-319-20239-6_8

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-20239-6_8

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-20238-9

  • Online ISBN: 978-3-319-20239-6

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation