Skip to main content
Log in

Simulation of a nonlinear Steklov eigenvalue problem using finite-element approximation

  • Published:
Computational Mathematics and Modeling Aims and scope Submit manuscript

Elliptic problems with parameters in the boundary conditions are called Steklov problems. With the tool of computational approximation (finite-element method), we estimate the solution of a nonlinear Steklov eigenvalue problem for a second-order, self-adjoint, elliptic differential problem. We discussed the behavior of the nonlinear problem with the help of computational results using Matlab.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. J. Canavati and A. Minsoni, “A discontinuous Steklov problem with an application to water waves.” J. Math. Anal. Appl., 69, 540–558 (1979).

    Article  MATH  MathSciNet  Google Scholar 

  2. D. B. Hinton and J. K. Shaw, “Differential operators with spectral parameter incompletely in the boundary conditions,” Funkcialaj Ekvacioj (Serio Internacia), 33, 363–385(1990).

    MATH  MathSciNet  Google Scholar 

  3. H. J. Ahn, “Vibration of a pendulum consisting of a bob suspended from a wire,” Quart. Appl. Math., 39, 109–117(1981).

    MATH  MathSciNet  Google Scholar 

  4. J. R. Kuttler and V. G. Sigillito, “Inequalities for membrane and Stekloff eigenvalues,” J. Math. Anal. Appl., 23, 148–160 (1968).

    Article  MATH  MathSciNet  Google Scholar 

  5. L. E. Payne, “Some isoperimetric inequalities for harmonic functions,” SIAM J. Math. Anal., 1, 354–359 (1970).

    Article  MATH  MathSciNet  Google Scholar 

  6. A. Ferrero, F. Gazzola, and T. Weth, “On a fourth order Steklov eigenvalue problem,” Analysis, 25, 315–332 (2005).

    Article  MATH  MathSciNet  Google Scholar 

  7. J. H. Bramble and J. E. Osborn, “Approximation of Steklov eigenvalues of non-selfadjoint second-order elliptic operators,” in: A. K. Aziz (ed.), Math. Foundations of the Finite-Element Method with Applications to PDE, Academic Press, New York (1972), pp. 387–408.

    Google Scholar 

  8. A. B. Andreev and A. H. Hristov, “On the variational aspects for elliptic problems with parameter on the boundary,” in: Recent Advances in Numerical Methods and Applications II, World Scientific, Singapore (1998), pp. 587–593.

    Google Scholar 

  9. I. Babuska and J. Osborn, Eigenvalue Problem. Handbook of Numerical Analysis Vol. II, North-Holland, Amsterdam (1991).

    Google Scholar 

  10. A. Bermúdez, R. Durán, and R. Rodríguez, “Finite-element solution of incompressible fluid–structure vibration problems,” Internat. J. Numer. Meth. Eng., 40, 1435–1448 (1997).

    Article  Google Scholar 

  11. A. Bermúdez, R. Rodríguez, and D. Santamarina, “A finite-element solution of an added mass formulation for coupled fluid – solid vibrations,” Numer. Math., 87, 201–227 (2000).

    Article  MATH  MathSciNet  Google Scholar 

  12. R. A. Adams, Sobolev Spaces, Academic Press, San Diego, CA (1978).

    Google Scholar 

  13. A. B. Andreev, “Isoparametric finite-element approximation of a Steklov eigenvalue problem,” IMA J. Numer. Anal., 24, 309–322 (2004).

    Article  MATH  MathSciNet  Google Scholar 

  14. W. Stekloff, “Sur les problèmes fondamentaux de la physique mathématique,” Ann. Sci. Ecol. Norm. Sup., 19, 455–490 (1902).

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Prashant Kumar.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kumar, P., Kumar, M. Simulation of a nonlinear Steklov eigenvalue problem using finite-element approximation. Comput Math Model 21, 109–116 (2010). https://doi.org/10.1007/s10598-010-9058-6

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10598-010-9058-6

Keywords

Navigation