Skip to main content

Advertisement

SpringerLink
Log in

On the Inequalities of Babuška–Aziz, Friedrichs and Horgan–Payne

  • Published:
Archive for Rational Mechanics and Analysis Aims and scope Submit manuscript

Abstract

The equivalence between the inequalities of Babuška–Aziz and Friedrichs for sufficiently smooth bounded domains in the plane was shown by Horgan and Payne 30 years ago. We prove that this equivalence, and the equality between the associated constants, is true without any regularity condition on the domain. For the Horgan–Payne inequality, which is an upper bound of the Friedrichs constant for plane star-shaped domains in terms of a geometric quantity known as the Horgan–Payne angle, we show that it is true for some classes of domains, but not for all bounded star-shaped domains. We prove a weaker inequality that is true in all cases.

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., Muschietti M.A.: Solutions of the divergence operator on John domains. Adv. Math. 206, 373–401 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  2. Babuška, I., Aziz, A.K.: Survey lectures on the mathematical foundations of the finite element method. In: The mathematical foundations of the finite element method with applications to partial differential equations (Proceedings of symposium on University of Maryland, Baltimore, MD., 1972), Academic Press, New York, pp. 1–359 (1972)

  3. Brezzi F.: On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers. Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge 8, 129–151 (1974)

    MATH  MathSciNet  Google Scholar 

  4. Cattabriga L.: Su un problema al contorno relativo al sistema di equazioni di Stokes. Rend. Sem. Mat. Univ. Padova 31, 308–340 (1961)

    MATH  MathSciNet  Google Scholar 

  5. Ciarlet, P.G., Ciarlet, Jr., P.: Another approach to linearized elasticity and a new proof of Korn’s inequality. Math. Models Methods Appl. Sci. 15, 259–271 (2005)

  6. Crouzeix, M.: On an operator related to the convergence of Uzawa’s algorithm for the Stokes equation. In: Computational science for the 21st century (Eds. M.-O. Bristeau, G. Etgen, W. Fitzgibbon, J. Lions, J. Périaux and M. Wheeler) Chichester, Wiley, pp. 242–249 (1997)

  7. Dauge, M., Bernardi, C., Costabel, M., Girault, V.: On Friedrichs constant and Horgan–Payne angle for LBB condition, In: Twelfth International Conference Zaragoza-Pau on Mathematics, Monogr Mat García Galdeano, Prensas Univ. Zaragoza, vol. 39, pp. 87–100, Zaragoza (2014)

  8. Dobrowolski M.: On the LBB condition in the numerical analysis of the Stokes equations. Appl. Numer. Math. 54, 314–323 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  9. Duran, R., Muschietti, M.-A., Russ, E., Tchamitchian, P.: Divergence operator and Poincaré inequalities on arbitrary bounded domains. Complex Var. Elliptic Equ. 55, 795–816 (2010)

  10. Durán, R.G.: An elementary proof of the continuity from \({L_0^2(\Omega)}\) to \({H^1_0(\Omega)^n}\) of Bogovskii’s right inverse of the divergence. Revista de la Unión Matemática Argentina 53, 59–78 (2012)

  11. Friedrichs K.: On certain inequalities and characteristic value problems for analytic functions and for functions of two variables. Trans. Amer. Math. Soc. 41, 321–364 (1937)

    Article  MathSciNet  Google Scholar 

  12. Horgan, C.O., Payne, L.E.: On inequalities of Korn, Friedrichs and Babuška–Aziz. Arch. Ration. Mech. Anal. 82, 165–179 (1983)

  13. Ladyzhenskaya O., Solonnikov V.: Some problems of vector analysis and generalized formulations of boundary-value problems for the Navier–Stokes equations. J. Sov. Math. 8, 257–286 (1978)

    Article  MATH  Google Scholar 

  14. Magenes, E., Stampacchia, G.: I problemi al contorno per le equazioni differenziali di tipo ellittico. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze 12, 247–358 (1958)

  15. Malkus D.S.: Eigenproblems associated with the discrete LBB condition for incompressible finite elements. Int. J. Eng. Sci. 19, 1299–1310 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  16. Maz’ya, V.: Sobolev spaces with applications to elliptic partial differential equations, vol. 342 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer, Heidelberg, augmented ed. (2011)

  17. Miyazaki Y.: New proofs of the trace theorem of Sobolev spaces. Proc. Jpn Acad. Ser. A Math. Sci. 84, 112–116 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  18. Nečas, J.: Les méthodes directes en théorie des équations elliptiques. Masson-Academia, Paris-Prague, 1967

  19. Oden J.T., Kikuchi N., Song Y.J.: Penalty-finite element methods for the analysis of Stokesian flows. Comput. Methods Appl. Mech. Eng. 31, 297–329 (1982)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  20. Stoyan G.: Towards discrete Velte decompositions and narrow bounds for inf–sup constants. Comput. Math. Appl. 38, 243–261 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  21. Stoyan G.: Iterative Stokes solvers in the harmonic Velte subspace. Computing 67, 13–33 (2001)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. IRMAR UMR 6625 du CNRS, Université de Rennes 1, Campus de Beaulieu, 35042, Rennes Cedex, France

    Martin Costabel & Monique Dauge

Authors
  1. Martin Costabel
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Monique Dauge
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Martin Costabel.

Additional information

Communicated by G. Friesecke

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Costabel, M., Dauge, M. On the Inequalities of Babuška–Aziz, Friedrichs and Horgan–Payne. Arch Rational Mech Anal 217, 873–898 (2015). https://doi.org/10.1007/s00205-015-0845-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00205-015-0845-2

Keywords

Search

Navigation