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Zh. Vychisl. Mat. Mat. Fiz., 2007, Volume 47, Number 5, Pages 835–866 (Mi zvmmf292)  

This article is cited in 7 scientific papers (total in 7 papers)

Approximation of systems of singularly perturbed elliptic reaction-diffusion equations with two parameters

G. I. Shishkin

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

Abstract: In a rectangle, the Dirichlet problem for a system of two singularly perturbed elliptic reaction-diffusion equations is considered. The higher order derivatives of the $i$th equation are multiplied by the perturbation parameter $\varepsilon_i^2$ ($i=1,2$). The parameters $\varepsilon_i$ take arbitrary values in the half-open interval $(0,1]$. When the vector parameter $\boldsymbol\varepsilon=(\varepsilon_1, \varepsilon_2)$ vanishes, the system of elliptic equations degenerates into a system of algebraic equations. When the components $\varepsilon_1$ and (or) $\varepsilon_2$ tend to zero, a double boundary layer with the characteristic width $\varepsilon_1$ and $\varepsilon_2$ appears in the vicinity of the boundary. Using the grid refinement technique and the classical finite difference approximations of the boundary value problem, special difference schemes that converge $\boldsymbol\varepsilon$-uniformly at the rate of $O(N^{-2}\ln^2N)$ are constructed, where $N=\min_sN_s$, $N_s+1$ is the number of mesh points on the axis $x_s$.

Key words: singularly perturbed elliptic equation, system of reaction-diffusion equations with two parameters, finite difference method, double boundary layer, rate of convergence at a difference scheme, $\varepsilon$-uniform convergence.

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English version:
Computational Mathematics and Mathematical Physics, 2007, 47:5, 797–828

Bibliographic databases:

UDC: 519.633
Received: 06.12.2006

Citation: G. I. Shishkin, “Approximation of systems of singularly perturbed elliptic reaction-diffusion equations with two parameters”, Zh. Vychisl. Mat. Mat. Fiz., 47:5 (2007), 835–866; Comput. Math. Math. Phys., 47:5 (2007), 797–828

Citation in format AMSBIB
\Bibitem{Shi07}
\by G.~I.~Shishkin
\paper Approximation of systems of singularly perturbed elliptic reaction-diffusion equations with two parameters
\jour Zh. Vychisl. Mat. Mat. Fiz.
\yr 2007
\vol 47
\issue 5
\pages 835--866
\mathnet{http://mi.mathnet.ru/zvmmf292}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=2378662}
\transl
\jour Comput. Math. Math. Phys.
\yr 2007
\vol 47
\issue 5
\pages 797--828
\crossref{https://doi.org/10.1134/S0965542507050077}
\scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-34249732999}


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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. 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
    2. Shishkina L., Shishkin G., “Robust numerical method for a system of singularly perturbed parabolic reaction-diffusion equations on a rectangle”, Math. Model. Anal., 13:2 (2008), 251–261  crossref  mathscinet  zmath  isi  elib  scopus
    3. 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
    4. Shishkina L., Shishkin G., “Conservative numerical method for a system of semilinear singularly perturbed parabolic reaction-diffusion equations”, Math. Model. Anal., 14:2 (2009), 211–228  crossref  mathscinet  zmath  isi  elib  scopus
    5. Kadalbajoo M.K., Gupta V., “A brief survey on numerical methods for solving singularly perturbed problems”, Applied Mathematics and Computation, 217:8 (2010), 3641–3716  crossref  mathscinet  zmath  isi  scopus
    6. Clavero C., Jorge J.C., “An Efficient Numerical Method For Singularly Perturbed Time Dependent Parabolic 2D Convection-Diffusion Systems”, J. Comput. Appl. Math., 354 (2019), 431–444  crossref  mathscinet  isi  scopus
    7. Clavero C. Gracia J.L., “Uniformly Convergent Additive Schemes For 2D Singularly Perturbed Parabolic Systems of Reaction-Diffusion Type”, Numer. Algorithms, 80:4 (2019), 1097–1120  crossref  mathscinet  zmath  isi  scopus
  • Журнал вычислительной математики и математической физики Computational Mathematics and Mathematical Physics
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