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 Zh. Vychisl. Mat. Mat. Fiz., 2008, Volume 48, Number 6, Pages 1014–1033 (Mi zvmmf4577)

Grid approximation of a parabolic convection-diffusion equation on a priori adapted grids: $\varepsilon$-uniformly convergent schemes

G. I. Shishkin

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

Abstract: The boundary value problem for a singularly perturbed parabolic convection-diffusion equation is considered. A finite difference scheme on a priori (sequentially) adapted grids is constructed and its convergence is examined. The construction of the scheme on a priori adapted grids is based on a majorant of the singular component of the grid solution that makes it possible to a priori find a subdomain in which the grid solution should be further refined given the perturbation parameter $\varepsilon$, the size of the uniform mesh in $x$, the desired accuracy of the grid solution, and the prescribed number of iterations $K$ used to refine the solution. In the subdomains where the solution is refined, the grid problems are solved on uniform grids. The error of the solution thus constructed weakly depends on $\varepsilon$. The scheme converges almost $\varepsilon$-uniformly; namely, it converges under the condition $N^{-1}=o(\varepsilon^\nu)$, where $\nu=\nu(K)$ can be chosen arbitrarily small when $K$ is sufficiently large. If a piecewise uniform grid is used instead of a uniform one at the final $K$ th iteration, the difference scheme converges $\varepsilon$-uniformly. For this piecewise uniform grid, the ratio of the mesh sizes in $x$ on the parts of the mesh with a constant size (outside the boundary layer and inside it) is considerably less than that for the known $\varepsilon$-uniformly convergent schemes on piecewise uniform grids.

Key words: singular perturbations, convection-diffusion parabolic problem, piecewise uniform grid, a priori adapted grid, almost $\varepsilon$-uniform convergence, $\varepsilon$-uniform convergence.

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English version:
Computational Mathematics and Mathematical Physics, 2008, 48:6, 956–974

Bibliographic databases:

UDC: 519.633

Citation: G. I. Shishkin, “Grid approximation of a parabolic convection-diffusion equation on a priori adapted grids: $\varepsilon$-uniformly convergent schemes”, Zh. Vychisl. Mat. Mat. Fiz., 48:6 (2008), 1014–1033; Comput. Math. Math. Phys., 48:6 (2008), 956–974

Citation in format AMSBIB
\Bibitem{Shi08} \by G.~I.~Shishkin \paper Grid approximation of a~parabolic convection-diffusion equation on a~priori adapted grids: $\varepsilon$-uniformly convergent schemes \jour Zh. Vychisl. Mat. Mat. Fiz. \yr 2008 \vol 48 \issue 6 \pages 1014--1033 \mathnet{http://mi.mathnet.ru/zvmmf4577} \zmath{https://zbmath.org/?q=an:1164.35314} \transl \jour Comput. Math. Math. Phys. \yr 2008 \vol 48 \issue 6 \pages 956--974 \crossref{https://doi.org/10.1134/S0965542508060080} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000262334200008} \scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-45749148727} 

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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. I. A. Blatov, N. V. Dobrobog, “Conditional $\varepsilon$-uniform convergence of adaptation algorithms in the finite element method for singularly perturbed problems”, Comput. Math. Math. Phys., 50:9 (2010), 1476–1493
2. Shishkin G.I., “Difference scheme of the solution decomposition method for a singularly perturbed parabolic reaction-diffusion equation”, Russian J. Numer. Anal. Math. Modelling, 25:3 (2010), 261–278
3. A. P. Vlasyuk, P. N. Martynyuk, “Kontaktnyi razmyv i filtratsionnaya konsolidatsiya gruntov v usloviyakh teplo-soleperenosa”, Matem. modelirovanie, 24:11 (2012), 97–112
4. I. A. Blatov, E. V. Kitaeva, “Convergence of the adapting grid method of Bakhvalov's type for singularly perturbed boundary value problems”, Num. Anal. Appl., 9:1 (2016), 34–44
5. I. A. Blatov, N. V. Dobrobog, E. V. Kitaeva, “Conditional $\varepsilon$-uniform boundedness of Galerkin projectors and convergence of an adaptive mesh method as applied to singularly perturbed boundary value problems”, Comput. Math. Math. Phys., 56:7 (2016), 1293–1304
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