Geodesic
An almost fourth order uniformly convergent scheme for reaction-diffusion problems on a piecewise uniform grid
Čebyševskij sbornik, Tome 12 (2011) no. 1, pp. 51-59.

Voir la notice de l'article provenant de la source Math-Net.Ru

For the problem: $-\varepsilon y^{\prime\prime}(x) + p(x) y(x) = f(x),\;\: x \in D, \;\: y(0)=\alpha_{0},\, y(1)=\alpha_{1}$ the spline difference schemes on a piecewise mesh having the second order of uniform convergence are given. Also, in this paper is presented a construction of an uniformly convergent scheme with an almost fourth order of uniform convergence on a Shishkin mesh. 
@article{CHEB_2011_12_1_a3,
     author = {Vanja Vukoslav\v{c}evi\'c},
     title = {An almost fourth order uniformly convergent scheme for reaction-diffusion problems on a piecewise uniform grid},
     journal = {\v{C}eby\v{s}evskij sbornik},
     pages = {51--59},
     publisher = {mathdoc},
     volume = {12},
     number = {1},
     year = {2011},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CHEB_2011_12_1_a3/}
}
TY  - JOUR
AU  - Vanja Vukoslavčević
TI  - An almost fourth order uniformly convergent scheme for reaction-diffusion problems on a piecewise uniform grid
JO  - Čebyševskij sbornik
PY  - 2011
SP  - 51
EP  - 59
VL  - 12
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/CHEB_2011_12_1_a3/
LA  - en
ID  - CHEB_2011_12_1_a3
ER  - 
%0 Journal Article
%A Vanja Vukoslavčević
%T An almost fourth order uniformly convergent scheme for reaction-diffusion problems on a piecewise uniform grid
%J Čebyševskij sbornik
%D 2011
%P 51-59
%V 12
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/CHEB_2011_12_1_a3/
%G en
%F CHEB_2011_12_1_a3
Vanja Vukoslavčević. An almost fourth order uniformly convergent scheme for reaction-diffusion problems on a piecewise uniform grid. Čebyševskij sbornik, Tome 12 (2011) no. 1, pp. 51-59. http://geodesic.mathdoc.fr/item/CHEB_2011_12_1_a3/

[1] E. O'Riordan, M. Stynes, “A Uniformly Accurate Finite-Element Method for a Singularly Perturbed One-Dimension Reaction-Diffusion Problem”, Math. Comp., 47:176 (1986), 555–579 | MR

[2] R. B. Kellog, A. Tsan, “Analysis of Some Difference approximations for Singular Perturbation Problem Without Turning Points”, Math. Comp., 32:44 (1978), 1025–1039 | DOI | MR | Zbl

[3] K. Surla, D. Herceg, Lj. Cvetkovic, “A family of exponentially spline difference schemes”, Univ. u place N. Sadu, Zb. Rad. Prirod. Mat. Fak. Ser. Mat., 19:2 (1991), 12–23 | MR

[4] K. Surla, V. Vukoslavcevic, “A spline difference scheme for boundary value problem with small parameter”, Univ. u place N. Sadu, Zb. Rad. Prirod. Mat. Fak. Ser. Mat., 25:2 (1995), 159–168 | MR | Zbl

[5] K. Surla, V. Vukoslavcevic, “Global approximations of the solution of the boundary value problem with small parameter”, Proceedings of the IX-th Conference on Applied Mathematics, Instit. Of Math., N. Sad, 1995, 133–141 | MR

[6] Z. Uzelac, Spline difference schemes for solving singularly perturbed problem, Ph. D. Thesis, University of Novi Sad, 1989 | Zbl

[7] V. Vukoslavcevic, Stability of difference methods for solving differential equations, Ph. D. Thesis, University of Novi Sad, 1999

[8] V. Vukoslavcevic, K. Surla, S. Rapajic, “Some uniformly convergent schemes on Shishkin mesh”, N. Sad J. Math., 28:2 (1998), 33–42 | MR | Zbl

[9] G. Sun, M. Stynes, A uniformly convergent Method for a Singularly Perturbed Semilinear Reaction-Diffusion Problem with Nonique Solutions, Preprint, University College Cork, 1993, 24 pp.

[10] Shishkin G. I., A Discrete Approximation of Singularly Perturbed Elliptic and Parabolic Equations, Russian Academy of Sciences, Ural Section, Ekaterinburg, 1992

[11] Hegarty F. A., Miller H. J. J., O'Riordan E., “Uniformly second order difference scheme for singularly perturbed problem”, Boundary and Interior Layer-Comutational and Asymptotic Methods, ed. J. J. Miller, Boole Press, Dublin, 1980, 301–305 | MR