Skip to main content
Log in

A Tailored Finite Point Method for a Singular Perturbation Problem on an Unbounded Domain

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

Abstract

In this paper, we propose a tailored-finite-point method for a kind of singular perturbation problems in unbounded domains. First, we use the artificial boundary method (Han in Frontiers and Prospects of Contemporary Applied Mathematics, [2005]) to reduce the original problem to a problem on bounded computational domain. Then we propose a new approach to construct a discrete scheme for the reduced problem, where our finite point method has been tailored to some particular properties or solutions of the problem. From the numerical results, we find that our new methods can achieve very high accuracy with very coarse mesh even for very small ε. In the contrast, the traditional finite element method does not get satisfactory numerical results with the same mesh.

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.

Institutional subscriptions

Similar content being viewed by others

References

  1. Berger, A.E., Han, H.D., Kellogg, R.B.: A priori estimates and analysis of a numerical method for a turning point problem. Math. Comp. 42, 465–492 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  2. Brayanov, I., Dimitrova, I.: Uniformly convergent high-order schemes for a 2D elliptic reaction-diffusion problem with anisotropic coefficients. Lect. Notes Comput. Sci. 2542, 395–402 (2003)

    Article  MathSciNet  Google Scholar 

  3. Cheng, M., Liu, G.R.: A novel finite point method for flow simulation. Int. J. Numer. Meth. Fluids 39, 1161–1178 (2002)

    Article  MATH  Google Scholar 

  4. Evans, L.C.: Partial Differential Equations. American Mathematical Society, Providence (1998)

    MATH  Google Scholar 

  5. Han, H.D.: The artificial boundary method – numerical solutions of partial differential equations on unbounded domains. In: Li, T., Zhang, P. (eds.) Frontiers and Prospects of Contemporary Applied Mathematics. Higher Education Press, World Scientific (2005)

    Google Scholar 

  6. Han, H.D., Bao, W.Z.: Error estimates for the finite element approximation of problems in unbounded domains. SIAM J. Numer. Anal. 37, 1101–1119 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  7. Hemker, P.W.: A singularly perturbed model problem for numerical computation. J. Comp. Appl. Math. 76, 277–285 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  8. Il’in, A.M.: Differencing scheme for a differential equation with a small parameter affecting the highest derivative. Math. Notes 6, 596–602 (1969)

    Google Scholar 

  9. Kellogg, R.B., Stynes, M.: A singularly perturbed convection-diffusion problem in a half-plane. Appl. Anal. 85, 1471–1485 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  10. Li, J., Navon, I.M.: Uniformly convergent finite element methods for singularly perturbed elliptic boundary value problems: convection-diffusion. Comput. Methods Appl. Mech. Eng. 162, 49–78 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  11. Lin, H., Atluri, S.N.: The meshless local Petrov-Galerkin (MLPG) method for solving incompressible Navier-Stokes equations. CMES 2, 117–142 (2001)

    MathSciNet  Google Scholar 

  12. Mendez a, B., Velazquez, A.: Finite point solver for the simulation of 2-D laminar incompressible unsteady flows. Comput. Methods Appl. Mech. Eng. 193, 825–848 (2004)

    Article  MATH  Google Scholar 

  13. Miller, J.J.H.: On the convergence, uniformly in ε, of difference schemes for a two-point boundary singular perturbation problem. In: Hernker, P.W., Miller, J.J.H. (eds.) Numerical Analysis of Singular Perturbation Problems, pp. 467–474. Academic, New York (1979)

    Google Scholar 

  14. Nassehi, V., Parvazinia, M., Khan, A.: Multiscale finite element modelling of flow through porous media with curved and contracting boundaries to evaluate different types of bubble functions. Commun. Comput. Phys. 2, 723–745 (2007)

    MathSciNet  Google Scholar 

  15. Oñate, E., Idelsohn, S., Zienkiewicz, O.C., Taylor, R.L.: A finite point method in computational mechanics. Applications to convective transport and fluid flow. Int. J. Numer. Methods Eng. 39, 3839–3866 (1996)

    Article  MATH  Google Scholar 

  16. Roos, H.-G., Stynes, M., Tobiska, L.: Numerical Methods for Singularly Perturbed Differential Equations. Springer, New York (1996)

    MATH  Google Scholar 

  17. Wang, M., Meng, X.R.: A robust finite element method for a 3-D elliptic singular perturbation problem. J. Comput. Math. 25, 631–644 (2007)

    MATH  MathSciNet  Google Scholar 

  18. Wang, M., Xu, J.C., Hu, Y.C.: Modified Morley element method for a fourth order elliptic singular perturbation problem. J. Comput. Math. 24, 113–120 (2006)

    MATH  MathSciNet  Google Scholar 

  19. Wesseling, P.: Uniform Convergence of Discretization Error for a Singular Perturbation Problem. Numer. Methods Partial Differ. Equ. 12, 657–671 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  20. Xie, Z.Q., Zhang, Z.M.: Superconvergence of DG method for one-dimensional singularly perturbed problems. J. Comput. Math. 25, 185–200 (2007)

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Zhongyi Huang.

Additional information

Han was supported by the NSFC Project No. 10471073.

Z. Huang was supported by the NSFC Projects No. 10301017, and 10676017, the National Basic Research Program of China under the grant 2005CB321701.

R.B. Kellogg was supported by the Boole Centre for Research in Informatics at National University of Ireland, Cork and by Science Foundation Ireland under the Basic Research Grant Programme 2004 (Grants 04/BR/M0055, 04/BR/M0055s1).

Rights and permissions

Reprints and permissions

About this article

Cite this article

Han, H., Huang, Z. & Kellogg, R.B. A Tailored Finite Point Method for a Singular Perturbation Problem on an Unbounded Domain. J Sci Comput 36, 243–261 (2008). https://doi.org/10.1007/s10915-008-9187-7

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10915-008-9187-7

Keywords

Navigation