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 Zh. Vychisl. Mat. Mat. Fiz., 2005, Volume 45, Number 1, Pages 110–125 (Mi zvmmf721)

Grid approximation of a singularly perturbed elliptic equation with convective terms in the presence of various boundary layers

G. I. Shishkin

Institute of Mathematics and Mechanics, Ural Division, Russian Academy of Sciences, ul. S. Kovalevskoi 16, Yekaterinburg, 620219, Russia

Abstract: The Dirichlet problem for a singularly perturbed elliptic convection-diffusion equation in a rectangle and in a vertical half-strip with a vector perturbation parameter $\varepsilon=(\varepsilon_1,\varepsilon_2)$ is considered. The higher derivatives of the equation and the first derivative with respect to the vertical coordinate include the parameters $\varepsilon_1$ and $\varepsilon_2$, respectively, which can take arbitrary values in the intervals $(0,1]$ and $[–1,1]$. For small values of $\varepsilon_1$, boundary layers appear in the neighborhood of various parts of the domain boundary. The type of these layers depends on the relation between $\varepsilon_1$ and $\varepsilon_2$: they can be regular, parabolic, or hyperbolic. Their characteristics also depend on the relation between $\varepsilon_1$ and $\varepsilon_2$. Using the special grid technique (these grids are condensing in the boundary layers), finite difference schemes are constructed that $\varepsilon$-uniformly converge in the maximum norm.

Key words: singularly perturbed problem for the elliptic equation, grid approximation, convergence, special grids.

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English version:
Computational Mathematics and Mathematical Physics, 2005, 45:1, 104–119

Bibliographic databases:
UDC: 519.632.4

Citation: G. I. Shishkin, “Grid approximation of a singularly perturbed elliptic equation with convective terms in the presence of various boundary layers”, Zh. Vychisl. Mat. Mat. Fiz., 45:1 (2005), 110–125; Comput. Math. Math. Phys., 45:1 (2005), 104–119

Citation in format AMSBIB
\Bibitem{Shi05} \by G.~I.~Shishkin \paper Grid approximation of a singularly perturbed elliptic equation with convective terms in the presence of various boundary layers \jour Zh. Vychisl. Mat. Mat. Fiz. \yr 2005 \vol 45 \issue 1 \pages 110--125 \mathnet{http://mi.mathnet.ru/zvmmf721} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2151047} \zmath{https://zbmath.org/?q=an:1114.65130} \elib{https://elibrary.ru/item.asp?id=9134020} \transl \jour Comput. Math. Math. Phys. \yr 2005 \vol 45 \issue 1 \pages 104--119 \elib{https://elibrary.ru/item.asp?id=13494603} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. Zh. Zh. Bai, L. A. Krukier, T. S. Martynova, “Two-step iterative methods for solving the stationary convection-diffusion equation with a small parameter at the highest derivative on a uniform grid”, Comput. Math. Math. Phys., 46:2 (2006), 282–293
2. G. I. Shishkin, “Approximation of systems of singularly perturbed elliptic reaction-diffusion equations with two parameters”, Comput. Math. Math. Phys., 47:5 (2007), 797–828
3. Teofanov L., Roos H.-G., “An elliptic singularly perturbed problem with two parameters. I. Solution decomposition”, J. Comput. Appl. Math., 206:2 (2007), 1082–1097
4. G. I. Shishkin, L. P. Shishkina, “Approximation of a system of singularly perturbed reaction-diffusion parabolic equations in a rectangle”, Comput. Math. Math. Phys., 48:4 (2008), 627–640
5. Shishkin G.I., “Grid approximation of singularly perturbed parabolic equations with moving boundary layers”, Math. Model. Anal., 13:3 (2008), 421–442
6. Teofanov L., Roos H.-G., “An elliptic singularly perturbed problem with two parameters. II. Robust finite element solution”, J. Comput. Appl. Math., 212:2 (2008), 374–389
7. Shishkin G.I., Shishkina L.P., “Approximation of a system of semilinear singularly perturbed parabolic reaction-diffusion equations on a vertical strip”, International Workshop on Multi-Rate Processes and Hysteresis, Journal of Physics Conference Series, 138, 2008
8. Teofanov L., Zarin H., “Superconvergence analysis of a finite element method for a two-parameter singularly perturbed problem”, BIT, 49:4 (2009), 743–765
9. Naughton A., Stynes M., “Regularity and derivative bounds for a convection-diffusion problem with Neumann boundary conditions on characteristic boundaries”, Z. Anal. Anwend., 29:2 (2010), 163–181
10. O'Riordan E., Pickett M.L., “A parameter-uniform numerical method for a singularly perturbed two parameter elliptic problem”, Adv Comput Math, 35:1 (2011), 57–82
11. Brdar M., Zarin H., Teofanov L., “A singularly perturbed problem with two parameters in two dimensions on graded meshes”, Comput. Math. Appl., 72:10 (2016), 2582–2603
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