Advertisement

Advances in Computational Mathematics

, Volume 24, Issue 1–4, pp 197–212 | Cite as

Uniform convergence analysis of finite difference approximations for singular perturbation problems on an adapted grid

  • Yanping Chen
Article

Abstract

A singularly perturbed two-point boundary value problem with an exponential boundary layer is solved numerically by using an adaptive grid method. The mesh is constructed adaptively by equidistributing a monitor function based on the arc-length of the approximated solutions. A first-order rate of convergence, independent of the perturbation parameter, is established by using the theory of the discrete Green's function. Unlike some previous analysis for the fully discretized approach, the present problem does not require the conservative form of the underlying boundary value problem.

Keywords

singular perturbation moving mesh rate of convergence error estimate 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    V.B. Andreev and I.A. Savin, On the convergence, uniform with respect to the small parameter, of A.A. Samarskii's monotone scheme and its modifications, Comput. Math. Math. Phys. 35 (1995) 581–591. MathSciNetGoogle Scholar
  2. [2]
    G.M. Beckett and J.A. Mackenzie, Convergence analysis of finite difference approximations on equidistributed grids to a singularly perturbed boundary value problem, Appl. Numer. Math. 35 (2000) 87–109. MathSciNetCrossRefGoogle Scholar
  3. [3]
    Y. Chen, Uniform pointwise convergence for a singularly perturbed problem using arc-length equidistribution, J. Comput. Appl. Math. 159(1) (2003) 25–34. zbMATHMathSciNetCrossRefGoogle Scholar
  4. [4]
    Y. Chen and J. Yang, Alternative convergence analysis for a convection–diffusion boundary value problem, submitted. Google Scholar
  5. [5]
    R.B. Kellogg and A. Tsan, Analysis of some difference approximations for a singular perturbation problem without turning points, Math. Comp. 32 (1978) 1025–1039. MathSciNetCrossRefGoogle Scholar
  6. [6]
    N. Kopteva, Maximum norm a posteriori error estimates for a one-dimensional convection–diffusion problem, SIAM J. Numer. Anal. 39 (2001) 423–441. zbMATHMathSciNetCrossRefGoogle Scholar
  7. [7]
    N. Kopteva and M. Stynes, A robust adaptive method for quasi-linear one-dimensional convection–diffusion problem, SIAM J. Numer. Anal. 39 (2001) 1446–1467. MathSciNetCrossRefGoogle Scholar
  8. [8]
    R. Li, T. Tang and P. Zhang, Moving mesh methods in multiple dimensions based on harmonic maps, J. Comput. Phys. 170 (2001) 562–588. MathSciNetCrossRefGoogle Scholar
  9. [9]
    T. Linss, Uniform pointwise convergence of finite difference schemes using grid equidistribution, Computing 66 (2001) 27–39. zbMATHMathSciNetCrossRefGoogle Scholar
  10. [10]
    T. Linss, H.-G. Roos and R. Vulanovic, Uniform pointwise convergence on Shishkin-type meshes for quasi-linear convection–diffusion problems, SIAM J. Numer. Anal. 38 (2000) 897–912. MathSciNetCrossRefGoogle Scholar
  11. [11]
    W.B. Liu and T. Tang, Error analysis for a Galerkin-spectral method with coordinate transformation for solving singularly perturbed problems, Appl. Numer. Math. 38 (2001) 315–345. MathSciNetCrossRefGoogle Scholar
  12. [12]
    J.A. Mackenzie, Uniform convergence analysis of an upwind finite-difference approximation of a convection–diffusion boundary value problem on an adaptive grid, IMA J. Numer. Anal. 19 (1999) 233–249. zbMATHMathSciNetCrossRefGoogle Scholar
  13. [13]
    L.S. Mulholland, Y. Qiu and D.M. Sloan, Solution of evolutionary partial differential equations using adaptive finite differences with pseudospectral post-processing, J. Comput. Phys. 131 (1997) 280–298. MathSciNetCrossRefGoogle Scholar
  14. [14]
    Y. Qiu and D.M. Sloan, Analysis of difference approximations to a singularly perturbed two-point boundary value problem on an adaptively generated grid, J. Comput. Appl. Math. 101 (1999) 1–25. MathSciNetCrossRefGoogle Scholar
  15. [15]
    Y. Qiu, D.M. Sloan and T. Tang, Numerical solution of a singularly perturbed two-point boundary value problem using equidistribution: Analysis of convergence, J. Comput. Appl. Math. 116 (2000) 121–143. MathSciNetCrossRefGoogle Scholar
  16. [16]
    H.-G. Roos, M. Stynes and L. Tobiska, Numerical Methods for Singularly Perturbed Differential Equations (Springer, Berlin, 1996). Google Scholar
  17. [17]
    G.I. Shishkin, Grid approximation of singularly perturbed elliptic and parabolic equations, Second Doctoral thesis, Keldysh Institute of Applied Mathematics, USSR Academy of Sciences, Moscow (1990). Google Scholar
  18. [18]
    D.R. Smart, Fixed-Point Theorems (Cambridge Univ. Press, Cambridge, 1974). Google Scholar
  19. [19]
    H.Z. Tang and T. Tang, Adaptive mesh methods for one- and two-dimensional hyperbolic conservation laws, SIAM J. Numer. Anal. 41 (2003) 487–515. MathSciNetCrossRefGoogle Scholar
  20. [20]
    T. Tang and M.R. Trummer, Boundary layer resolving pseudospectral methods for singular perturbation problems, SIAM J. Sci. Comput. 17 (1996) 430–438. MathSciNetCrossRefGoogle Scholar
  21. [21]
    R. Vulanovic, A priori meshes for singularly perturbed quasilinear two-point boundary value problems, IMA J. Numer. Anal. 21 (2001) 349–366. zbMATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer 2006

Authors and Affiliations

  1. 1.Department of MathematicsXiangtan UniversityHunanPeople's Republic of China

Personalised recommendations