Skip to main content
SpringerLink
Log in

Robust convergence of a compact fourth-order finite difference scheme for reaction–diffusion problems

  • Published:
Numerische Mathematik Aims and scope Submit manuscript

Abstract

We consider a singularly perturbed one-dimensional reaction–diffusion problem with strong layers. The problem is discretized using a compact fourth order finite difference scheme. Altough the discretization is not inverse monotone we are able to establish its maximum-norm stability and to prove its pointwise convergence on a Shishkin mesh. The convergence is uniform with respect to the perturbation parameter. Numerical experiments complement our theoretical results.

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. Bakhvalov N.S.: Towards optimization ofmethods for solving boundary value problems in the presence of boundary layers (In Russian). Zh. Vychisl. Mat. i Mat. Fiz. 9, 841–859 (1969)

    MATH  Google Scholar 

  2. Clavero C., Gracia J.L.: High order methods for elliptic and time dependent reaction–diffusion singularly perturbed problems. Appl. Math. Comput. 168, 1109–1127 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  3. Gracia J.L., Lisbona F., Clavero C.: High order ε-uniform methods for singularly perturbed reaction–diffusion problems. In: Vulkov, L., Waśniewski, J., Yalamov, P.(eds) Lecture Notes in Computer Science 1988, pp. 350–358. Springer, Berlin (2001)

    Google Scholar 

  4. Linß, T.: Layer-adapted meshes for convection-diffusion problems. Habilitation thesis, Technische Universität Dresden (2006)

  5. Linß T.: Maximum-norm error analysis of a non-monotone FEM for a singularly perturbed reaction–diffusion problem. BIT Numer. Math. 47(2), 379–391 (2007)

    Article  MATH  Google Scholar 

  6. Miller J.J.H., O’Riordan E., Shishkin G.I.: Fitted Numerical Methods for Singular Perturbation Problems Error Estimates in the Maximum Norm for Linear Problems in One and Two Dimensions. World Scientific, Singapore (1996)

    MATH  Google Scholar 

  7. Miller J.J.H., O’Riordan E., Shishkin G.I., Shishkina L.P.: Fitted mesh methods for problems with parabolic boundary layers. Math. Proc. R. Ir. Acad. 98(2), 173–190 (1998)

    MathSciNet  Google Scholar 

  8. Roos H.-G., Stynes M., Tobiska L.: Numerical Methods for Singularly Perturbed Differential Equations. Springer Series in Computational Mathematics, vol. 24. Springer, Berlin (1996)

    Google Scholar 

  9. Shishkin, G.I.: Grid Approximation of Singularly Perturbed Elliptic and Parabolic Equations (In Russian). Second doctorial thesis, Keldysh Institute, Moscow (1990)

Download references

Author information

Authors and Affiliations

  1. Institut für Numerische Mathematik, TU Dresden, 01219, Dresden, Germany

    Torsten Linß

Authors
  1. Torsten Linß
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Torsten Linß.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Linß, T. Robust convergence of a compact fourth-order finite difference scheme for reaction–diffusion problems. Numer. Math. 111, 239–249 (2008). https://doi.org/10.1007/s00211-008-0184-4

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00211-008-0184-4

Mathematics Subject Classification (2000)

Search

Navigation