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 Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki: Year: Volume: Issue: Page: Find

 Kazan. Gos. Univ. Uchen. Zap. Ser. Fiz.-Mat. Nauki, 2007, Volume 149, Book 4, Pages 146–172 (Mi uzku633)

Grid approximation of a singularly perturbed quasilinear parabolic convection-diffusion equation on a priori adapted meshes

G. I. Shishkin

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

Abstract: An initial-boundary value problem is considered for a quasilinear singularly perturbed parabolic convection-diffusion equation. For such a problem, a solution of a classical difference scheme on uniform grid converges at the rate $\mathcal O((\varepsilon+N^{-1})^{-1}N^{-1}+N_0^{-1})$, where $N+1$ and $N_0+1$ are the numbers of nodes in the meshes in $x$ and $t$ respectively; the scheme converges only under the condition $N^{-1}\ll\varepsilon$. In the present paper, nonlinear and linearized finite difference schemes are constructed on a priori sequentially adapted grids, and their convergence is studied. The construction of the schemes is carried out on the basis of a majorant to the singular component of the discrete solution on uniform grids that allows us to find a priori subdomains where the computed solution requires a further improvement. Such subdomain is defined by the perturbation parameter $\varepsilon$, the step-size of a uniform mesh in $x$, and also by the required accuracy of the grid solution and the prescribed number $K$ of iterations to refine the solution. The advantage of this approach consists in the uniform meshes used. The error of the discrete solution depends weakly on the parameter $\varepsilon$. The schemes that are constructed in the iterative process converge almost $\varepsilon$-uniformly, namely, under the condition $N^{-1}\ll\varepsilon^{\nu}$, where the value $\nu=\nu(K)$ can be chosen arbitrarily small for sufficiently large $K$.

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UDC: 519.633

Citation: G. I. Shishkin, “Grid approximation of a singularly perturbed quasilinear parabolic convection-diffusion equation on a priori adapted meshes”, Kazan. Gos. Univ. Uchen. Zap. Ser. Fiz.-Mat. Nauki, 149, no. 4, Kazan University, Kazan, 2007, 146–172

Citation in format AMSBIB
\Bibitem{Shi07} \by G.~I.~Shishkin \paper Grid approximation of a~singularly perturbed quasilinear parabolic convection-diffusion equation on {\it a~priori} adapted meshes \serial Kazan. Gos. Univ. Uchen. Zap. Ser. Fiz.-Mat. Nauki \yr 2007 \vol 149 \issue 4 \pages 146--172 \publ Kazan University \publaddr Kazan \mathnet{http://mi.mathnet.ru/uzku633}