Geodesic
Strong stability of a scheme on locally uniform meshes for a singularly perturbed ordinary differential convection–diffusion equation
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 52 (2012) no. 6, pp. 1010-1041 
G. I. Shishkin. Strong stability of a scheme on locally uniform meshes for a singularly perturbed ordinary differential convection–diffusion equation. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 52 (2012) no. 6, pp. 1010-1041. http://geodesic.mathdoc.fr/item/ZVMMF_2012_52_6_a6/
@article{ZVMMF_2012_52_6_a6,
     author = {G. I. Shishkin},
     title = {Strong stability of a scheme on locally uniform meshes for a singularly perturbed ordinary differential convection{\textendash}diffusion equation},
     journal = {\v{Z}urnal vy\v{c}islitelʹnoj matematiki i matemati\v{c}eskoj fiziki},
     pages = {1010--1041},
     year = {2012},
     volume = {52},
     number = {6},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZVMMF_2012_52_6_a6/}
}
TY  - JOUR
AU  - G. I. Shishkin
TI  - Strong stability of a scheme on locally uniform meshes for a singularly perturbed ordinary differential convection–diffusion equation
JO  - Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki
PY  - 2012
SP  - 1010
EP  - 1041
VL  - 52
IS  - 6
UR  - http://geodesic.mathdoc.fr/item/ZVMMF_2012_52_6_a6/
LA  - ru
ID  - ZVMMF_2012_52_6_a6
ER  - 
%0 Journal Article
%A G. I. Shishkin
%T Strong stability of a scheme on locally uniform meshes for a singularly perturbed ordinary differential convection–diffusion equation
%J Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki
%D 2012
%P 1010-1041
%V 52
%N 6
%U http://geodesic.mathdoc.fr/item/ZVMMF_2012_52_6_a6/
%G ru
%F ZVMMF_2012_52_6_a6

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

The Dirichlet problem for a singularly perturbed ordinary differential convection–diffusion equation with a small parameter $\varepsilon$ ($\varepsilon\in (0,1]$) multiplying the higher order derivative is considered. For the problem, a difference scheme on locally uniform meshes is constructed that converges in the maximum norm conditionally, i.e., depending on the relation between the parameter $\varepsilon$ and the value $N$ defining the number of nodes in the mesh used; in particular, the scheme converges almost $\varepsilon$-uniformly (i.e., its accuracy depends weakly on $\varepsilon$). The stability of the scheme with respect to perturbations in the data and its conditioning are analyzed. The scheme is constructed using classical monotone approximations of the boundary value problem on a priori adapted grids, which are uniform on subdomains where the solution is improved. The boundaries of these subdomains are determined by a majorant of the singular component of the discrete solution. On locally uniform meshes, the difference scheme converges at a rate of $O(\min[\varepsilon^{-1}N^{-K}\ln N, 1]+N^{-1}\ln N)$, where $K$ is a prescribed number of iterations for refining the discrete solution. The scheme converges almost $\varepsilon$-uniformly at a rate of $O(N^{-1}\ln N)$ if $N^{-1}\le \varepsilon^\nu$, where $\nu$ (the defect of $\varepsilon$-uniform convergence) determines the required number $K$ of iterations ($K=K(\nu)\sim \nu^{-1}$) and can be chosen arbitrarily small from the half-open interval $(0, 1]$. The condition number of the difference scheme satisfies the bound $\boldsymbol{\kappa}_P=O(\varepsilon^{-1/K}\ln^{1/K}\varepsilon^{-1}\delta^{-(K+1)/K})$ where $\delta$ is the accuracy of the solution of the scheme in the maximum norm in the absence of perturbations. For sufficiently large $K$, the scheme is almost $\varepsilon$-uniformly strongly stable.

[1] Shishkin G. I., “Aposteriorno adaptiruemye (po gradientu resheniya) setki v approksimatsii singulyarno vozmuschennykh uravnenii konvektsii-diffuzii”, Vychisl. tekhnologii, 6:1–2 (2001), 72–87 | MR | Zbl

[2] Shishkin G. I., “Approksimatsiya singulyarno vozmuschennykh uravnenii reaktsii-diffuzii na adaptivnykh setkakh”, Matem. modelirovanie, 13:3 (2001), 103–118 | MR | Zbl

[3] Shishkin G. I., Shishkina L. P., Hemker P. W., “A class of singularly perturbed convection-diffusion problems with a moving interior layer. A Posteriori Adaptive Mesh Technique”, Comp. Meth. Appl. Math., 4:1 (2004), 105–127 | MR | Zbl

[4] Shishkin G. I., “Ispolzovanie reshenii na vlozhennykh setkakh pri approksimatsii singulyarno vozmuschennogo parabolicheskogo uravneniya konvektsii-diffuzii na adaptiruyuschikh setkakh”, Zh. vychisl. matem. i matem. fiz., 46:9 (2006), 1617–1637 | MR

[5] Shishkin G. I., “A finite difference scheme on a priori adapted meshes for a singularly perturbed parabolic convection-diffusion equation”, Numer. Math. Theor. Meth. Appl., 1:2 (2008), 214–234 | MR | Zbl

[6] Shishkin G. I., “Raznostnaya skhema povyshennoi tochnosti na apriorno adaptiruyuschikhsya setkakh dlya singulyarno vozmuschennogo parabolicheskogo uravneniya konvektsii-diffuzii”, Zh. vychisl. matem. i matem. fiz., 51:10 (2011), 1816–1839 | MR | Zbl

[7] Shishkin G. I., Setochnye approksimatsii singulyarno vozmuschennykh ellipticheskikh i parabolicheskikh uravnenii, UrO RAN, Ekaterinburg, 1992

[8] Miller J. J. H., O'Riordan E., Shishkin G. I., Fitted Numerical Methods for Singular Perturbation Problems, World Scientific, Singapore, 1996 | MR

[9] Farrell P. A., Hegarty A. F., Miller J. J. H., O'Riordan E., Shishkin G. I., Robust Computational Techniques for Boundary Layers, Applied Mathematics, 16, Chapman and Hall/CRC Press, 2000 | MR | Zbl

[10] Roos H.-G., Stynes M., Tobiska L., Numerical Methods for Singularly Perturbed Differential Equations. Convection-Diffusion-Reaction and Flow Problems, Springer Series in Computational Mathematics, 24, Springer-Verlag, Berlin, 2008 | MR | Zbl

[11] Shishkin G. I., Shishkina L. P., Difference Methods for Singular Perturbation Problems, Ser. Monographs and Surveys in Pure and Applied Math., Chapman and Hall/CRC, 2009 | MR | Zbl

[12] Samarskii A. A., Teoriya raznostnykh skhem, Nauka, M., 1989 | MR

[13] Bakhvalov N. S., Chislennye metody, Nauka, M., 1973 | MR | Zbl

[14] Bakhvalov N. S., Zhidkov N. P., Kobelkov G. M., Chislennye metody, Laboratoriya bazovykh znanii, M., 2001