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 Trudy Inst. Mat. i Mekh. UrO RAN, 2012, Volume 18, Number 2, Pages 291–304 (Mi timm830)

Conditioning of a difference scheme of the solution decomposition method for a singularly perturbed convection-diffusion equation

G. I. Shishkin

Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences

Abstract: Conditioning of a difference scheme of the solution decomposition method is studied for a Dirichlet problem for a singularly perturbed ordinary differential convection-diffusion equation. In this scheme, we apply a decomposition of the discrete solution into the regular and singular components, which are solutions of discrete subproblems, i.e., classical difference approximations considered on uniform grids. The scheme converges $\varepsilon$-uniformly in the maximum norm at the rate $\mathcal O(N^{-1}\ln N)$; $\varepsilon$ is a perturbation parameter multiplying the high-order derivative in the equation, $\varepsilon\in(0,1]$, and $N+1$ is the number of nodes in the grids used. It is shown that the solution decomposition scheme, unlike the standard scheme on uniform grid, is $\varepsilon$-uniformly well conditioned and stable to perturbations in the data of the discrete problem; the conditioning number of the scheme is a value of order $\mathcal O(\delta^{-2}\ln\delta^{-1})$, where $\delta$ is the accuracy of the discrete solution.

Keywords: singularly perturbed boundary value problem, convection-diffusion equation, difference scheme of the solution decomposition method, uniform grids, $\varepsilon$-uniform convergence, maximum norm, $\varepsilon$-uniform stability of the scheme, $\varepsilon$-uniform well conditioning of the scheme.

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Citation: G. I. Shishkin, “Conditioning of a difference scheme of the solution decomposition method for a singularly perturbed convection-diffusion equation”, Trudy Inst. Mat. i Mekh. UrO RAN, 18, no. 2, 2012, 291–304

Citation in format AMSBIB
\Bibitem{Shi12} \by G.~I.~Shishkin \paper Conditioning of a~difference scheme of the solution decomposition method for a~singularly perturbed convection-diffusion equation \serial Trudy Inst. Mat. i Mekh. UrO RAN \yr 2012 \vol 18 \issue 2 \pages 291--304 \mathnet{http://mi.mathnet.ru/timm830} \elib{https://elibrary.ru/item.asp?id=17736208} 

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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. G. I. Shishkin, “Conditioning and stability of finite difference schemes on uniform meshes for a singularly perturbed parabolic convection-diffusion equation”, Comput. Math. Math. Phys., 53:4 (2013), 431–454
2. Shishkin G.I., “Data Perturbation Stability of Difference Schemes on Uniform Grids for a Singularly Perturbed Convection-Diffusion Equation”, Russ. J. Numer. Anal. Math. Model, 28:4 (2013), 381–417
3. G. I. Shishkin, L. P. Shishkina, “Ustoichivaya standartnaya raznostnaya skhema dlya singulyarno vozmuschennogo uravneniya konvektsii-diffuzii pri kompyuternykh vozmuscheniyakh”, Tr. IMM UrO RAN, 20, no. 1, 2014, 322–333
4. G. I. Shishkin, “Computer difference scheme for a singularly perturbed convection-diffusion equation”, Comput. Math. Math. Phys., 54:8 (2014), 1221–1233
5. G. I. Shishkin, “Difference scheme for a singularly perturbed parabolic convection–diffusion equation in the presence of perturbations”, Comput. Math. Math. Phys., 55:11 (2015), 1842–1856
6. G. I. Shishkin, “Computer difference scheme for a singularly perturbed elliptic convection-diffusion equation in the presence of perturbations”, Comput. Math. Math. Phys., 57:5 (2017), 815–832
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