Skip to main content
Log in

Numerical Approximation of Second-Order Elliptic Problems in Unbounded Domains

  • Published:
Journal of Scientific Computing Aims and scope Submit manuscript

Abstract

This paper deals with the numerical resolution of elliptic problems in unbounded domains using inverted finite elements. In opposition to conventional approaches which are based on the truncation of the domain, the suggested method keeps the domain unbounded and is based on a description of the asymptotic behavior in an appropriate functional framework. The method and its mathematical properties are presented first, and some computational examples are carried out. The obtained numerical results demonstrate the efficiency of the method.

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

Access this article

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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

Instant access to the full article PDF.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8

Similar content being viewed by others

References

  1. Alliot, F., Amrouche, C.: Problème de Stokes dans \({ \rm R}^n\) et espaces de Sobolev avec poids. C. R. Acad. Sci. Paris Sér. I Math. 325(12), 1247–1252 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  2. Amrouche, C., Girault, V., Giroire, J.: Weighted Sobolev spaces for Laplace’s equation in \({\rm R}^n\). J. Math. Pures Appl. (9) 73(6), 579–606 (1994)

    MATH  MathSciNet  Google Scholar 

  3. Arar, N., Boulmezaoud, T.Z.: Eigenfunctions of a weighted Laplace operator in the whole space. J. Math. Anal. Appl. 400(1), 161–173 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  4. Bayliss, A., Turkel, E.: Radiation boundary conditions for wave-like equations. Commun. Pure Appl. Math. 33, 707–725 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  5. Beer, G., Smith, I.M., Duenser, C.: The boundary element method with programming: for engineers and scientists, vol. 72. Springer, Berlin (2008)

  6. Bettess, P., Zienkiewicz, O.C.: Diffraction and refraction of surface waves using finite and infinite elements. Int. J. Numer. Methods Eng. 11(8), 1271–1290 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  7. Boulmezaoud, T.Z.: On the Stokes system and on the biharmonic equation in the half-space: an approach via weighted Sobolev spaces. Math. Methods Appl. Sci. 25(5), 373–398 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  8. Boulmezaoud, T.Z.: On the Laplace operator and on the vector potential problems in the half-space: an approach using weighted spaces. Math. Methods Appl. Sci. 26(8), 633–669 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  9. Boulmezaoud, T.Z.: Inverted finite elements: a new method for solving elliptic problems in unbounded domains. M2AN Math. Model. Numer. Anal. 39(1), 109–145 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  10. Boulmezaoud, T.Z., Babatin, M.M., Mziou, S.: Approximation of singular and radial elliptic problems in unbounded domains (in preparation)

  11. Boulmezaoud, T.Z., Medjden, M.: Weighted \(L^p\) theory of the Stokes and the bilaplacian operators in the half-space. J. Math. Anal. Appl. 342(1), 220–245 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  12. Boulmezaoud, T.Z., Razafison, U.: On the steady Oseen problem in the whole space. Hiroshima Math. J. 35(3), 371–401 (2005)

    MATH  MathSciNet  Google Scholar 

  13. Brebbia, C.A., Telles, J.C.F., Wrobel, L.C.: Boundary Element Techniques. Springer, Berlin (1984)

    Book  MATH  Google Scholar 

  14. Burnett, D.S.: A three-dimensional acoustic infinite element based on a prolate spheroidal multipole expansion. J. Acoust. Soc. Am. 96(5, part 1), 2798–2816 (1994)

    Google Scholar 

  15. Colton, D.L., Kress, R.: Integral Equation Methods in Scattering Theory. Pure and Applied Mathematics. Wiley, New York (1983)

    Google Scholar 

  16. Enquist, B., Majda, A.: Radiation boundary conditions for acoustic and elastic wave calculations. Comm. Pure Appl. Math. 32, 313–357 (1979)

    Article  Google Scholar 

  17. Gerdes, K.: A review of infinite element methods for exterior Helmholtz problems. Finite elements for wave problems. J. Comput. Acoust. 8(1), 43–62 (2000)

    Article  MathSciNet  Google Scholar 

  18. Gerdes, K., Demkowicz, L.: Solution of \(3\)D-Laplace and Helmholtz equations in exterior domains using \(hp\)-infinite elements. Comput. Methods Appl. Mech. Eng. 137(3–4), 239–273 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  19. Giroire, J.: Etude de quelques problèmes aux limites extérieurs et résolution par équations intégrales. Thèse de Doctorat d’Etat. Université Pierre et Marie Curie, Paris (1987)

  20. Hanouzet, B.: Espaces de Sobolev avec poids application au problème de Dirichlet dans un demi espace. Rend. Sem. Mat. Univ. Padova 46, 227–272 (1971)

    MathSciNet  Google Scholar 

  21. Ihlenburg, F.: Finite Element Analysis of Acoustic Scattering, vol. 132 of Applied Mathematical Sciences. Springer, New York (1998)

    Book  Google Scholar 

  22. Kondratev, V.A.: Boundary value problems for elliptic equations in domains with conical or angular points. Trans. Moscow Math. Soc. 16, 227–313 (1967)

    Google Scholar 

  23. Kufner, A.: Weighted Sobolev Spaces. A Wiley-Interscience Publication. Wiley, New York (1985)

    Google Scholar 

  24. Toselli, A.: A new family of exponential infinite elements for the analysis of lossless electromagnetic waveguides. Comput. Methods Appl. Mech. Eng. 140, 221–235 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  25. Tsynkov, S.V.: Numerical solution of problems on unbounded domains. A review. Appl. Numer. Math. 27(4), 465–532, Absorbing boundary conditions (1998)

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Tahar Z. Boulmezaoud.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Boulmezaoud, T.Z., Mziou, S. & Boudjedaa, T. Numerical Approximation of Second-Order Elliptic Problems in Unbounded Domains. J Sci Comput 60, 295–312 (2014). https://doi.org/10.1007/s10915-013-9798-5

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10915-013-9798-5

Keywords

Mathematics Subject Classification

Navigation