Skip to main content
SpringerLink
Log in

Interpolation error estimates in W 1,p for degenerate Q 1 isoparametric elements

  • Published:
Numerische Mathematik Aims and scope Submit manuscript

Abstract

Optimal order error estimates in H 1, for the Q 1 isoparametric interpolation were obtained in Acosta and Durán (SIAM J Numer Anal37, 18–36, 1999) for a very general class of degenerate convex quadrilateral elements. In this work we show that the same conlusions are valid in W 1,p for 1≤ p < 3 and we give a counterexample for the case p  ≥  3, showing that the result cannot be generalized for more regular functions. Despite this fact, we show that optimal order error estimates are valid for any p  ≥  1, keeping the interior angles of the element bounded away from 0 and π, independently of the aspect ratio. We also show that the restriction on the maximum angle is sharp for p  ≥  3.

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. Acosta G., Durán R.G. (1999). The maximum angle condition for mixed and nonconforming elements: Application to the Stokes equations. SIAM J. Numer. Anal. 37, 18–36

    Article  MathSciNet  MATH  Google Scholar 

  2. Acosta G., Durán R.G. (2000). Error estimates for Q 1 isoparametric elements satisfying a weak angle condition. SIAM J. Numer. Anal. 38, 1073–1088

    Article  MathSciNet  MATH  Google Scholar 

  3. Acosta G., Durán R.G. (2004). An optimal Poincaré inequality in L 1 for convex domains. Proc. Am. Math. Soc. 132, 195–202

    Article  MATH  Google Scholar 

  4. Apel, T.: Anisotropic finite elements: local estimates and applications. Advances in numerical mathematics. B. G. Teubner, Stuttgart, Leipzig (1999)

  5. Babuška I., Aziz A.K. (1976). On the angle condition in the finite element method. SIAM J. Numer. Anal. 13, 214–226

    Article  MathSciNet  MATH  Google Scholar 

  6. Brenner S.C., Scott R.L.: The mathematical theory of finite element methods 2nd edn. Text in applied mathematics 15. Springer,Berlin Heidelberg Newyork (2002)

  7. Ciarlet P.G., Raviart P.A. (1972). Interpolation theory over curved elements, with applications to finite elements methods. Comput. Methods Appl. Mech. Eng. 1, 217–249

    Article  MathSciNet  MATH  Google Scholar 

  8. Jamet P. (1976). Estimations d’erreur pour des éléments finis droits presque dégénérés. RAIRO Anal. Numér. 10, 46-61

    MathSciNet  Google Scholar 

  9. Jamet P. (1977). Estimation of the interpolation error for quadrilateral finite elements which can degenerate into triangles. SIAM J. Numer. Anal. 14, 925–930

    Article  MathSciNet  MATH  Google Scholar 

  10. Ming P., Shi Z.C. (2002). Quadrilateral mesh revisited. Comput. Methods Appl. Mech. Eng. 191, 5671–5682

    Article  MathSciNet  MATH  Google Scholar 

  11. Payne L. E., Weinberger H.F. (1960). An optimal Poincaré inequality for convex domains. Arch. Rat. Mech. Anal. 5, 286–292

    Article  Google Scholar 

  12. Verfürth R. (1999). Error estimates for some quasi-interpolation operator. AN Math. Model. Numer. Anal. 33, 695–713

    Article  MathSciNet  MATH  Google Scholar 

  13. Zenisek A., Vanmaele M. (1995). The interpolation theorem for narrow quadrilateral isoparametric finite elements. Numer. Math. 72, 123–141

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Universidad Nacional De General Sarmiento, J. M. Gutierrez 1150 Los Polvorines, B1613GSX, Buenos Aires, Argentina

    Gabriel Acosta & Gabriel Monzón

Authors
  1. Gabriel Acosta
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Gabriel Monzón
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Gabriel Acosta.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Acosta, G., Monzón, G. Interpolation error estimates in W 1,p for degenerate Q 1 isoparametric elements. Numer. Math. 104, 129–150 (2006). https://doi.org/10.1007/s00211-006-0018-1

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00211-006-0018-1

AMS Subject Classification

Search

Navigation