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

This article is cited in 11 scientific papers (total in 11 papers)

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
Received: 05.04.2004

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
\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
\jour Comput. Math. Math. Phys.
\yr 2005
\vol 45
\issue 1
\pages 104--119

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    Citing articles on Google Scholar: Russian citations, English citations
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    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  mathnet  crossref  mathscinet  zmath
    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  mathnet  crossref  mathscinet
    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  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    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  mathnet  crossref  mathscinet  zmath  isi
    5. Shishkin G.I., “Grid approximation of singularly perturbed parabolic equations with moving boundary layers”, Math. Model. Anal., 13:3 (2008), 421–442  crossref  mathscinet  zmath  isi  elib  scopus
    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  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    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  crossref  isi  scopus
    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  crossref  mathscinet  zmath  isi  elib  scopus
    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  crossref  mathscinet  zmath  isi  elib  scopus
    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  crossref  mathscinet  zmath  isi  scopus
    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  crossref  mathscinet  zmath  isi  elib  scopus
  • Журнал вычислительной математики и математической физики Computational Mathematics and Mathematical Physics
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