Advertisement

Numerische Mathematik

, Volume 111, Issue 2, pp 239–249 | Cite as

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

  • Torsten Linß
Article

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.

Mathematics Subject Classification (2000)

65L10 65L12 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 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)MATHGoogle Scholar
  2. 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)MATHCrossRefMathSciNetGoogle Scholar
  3. 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. 4.
    Linß, T.: Layer-adapted meshes for convection-diffusion problems. Habilitation thesis, Technische Universität Dresden (2006)Google Scholar
  5. 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)MATHCrossRefGoogle Scholar
  6. 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)MATHGoogle Scholar
  7. 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)MathSciNetGoogle Scholar
  8. 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. 9.
    Shishkin, G.I.: Grid Approximation of Singularly Perturbed Elliptic and Parabolic Equations (In Russian). Second doctorial thesis, Keldysh Institute, Moscow (1990)Google Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Institut für Numerische MathematikDresdenGermany

Personalised recommendations