Zhurnal Vychislitel'noi Matematiki i Matematicheskoi Fiziki
 RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
 General information Latest issue Archive Impact factor Search papers Search references RSS Latest issue Current issues Archive issues What is RSS

 Zh. Vychisl. Mat. Mat. Fiz.: Year: Volume: Issue: Page: Find

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

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.

Full text: PDF file (3020 kB)
References: PDF file   HTML file

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

Bibliographic databases:

UDC: 519.633

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{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-34249732999} 

• http://mi.mathnet.ru/eng/zvmmf292
• http://mi.mathnet.ru/eng/zvmmf/v47/i5/p835

 SHARE:

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. 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
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
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
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
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
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
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
•  Number of views: This page: 1400 Full text: 260 References: 50 First page: 1