
This article is cited in 3 scientific papers (total in 3 papers)
Grid approximation of the domain and solution decomposition method with improved convergence rate for singularly perturbed elliptic equations in domains with characteristic boundaries
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 singularly perturbed elliptic equations with convection terms is considered in the case when the characteristics of the reduced equations are parallel to the rectangle sides. The higher order derivatives in the equations are multiplied by a perturbation parameter $\tilde\varepsilon=\varepsilon^2$ that can take arbitrary values in the halfopen interval $(0,1]$. For such convection–diffusion problems, the order of the $\varepsilon$uniform convergence (in the maximum norm) of the wellknown special schemes on piecewise uniform grids is not higher than unity (with respect to the variable along the flow). In this paper, a scheme on piecewise uniform grids is constructed that converges $\varepsilon$uniformly at the rate of $O(N^{2}\ln^2N)$, where $N$ specifies the number of mesh points with respect to each variable. When is not very small (compared to the effective mesh size in the direction of the convective flow) this scheme approximates the equation using central difference derivatives. For small $\tilde\varepsilon$, a domain decomposition method is used; more precisely, the problem is considered separately in a neighborhood of the outflow part of the domain boundary and outside this neighborhood. In the neighborhood of the outflow part of the boundary, central difference derivatives are used. Outside this neighborhood, the solution is decomposed. The regular part of the solution and the parabolic boundary layer are found by solving the corresponding problems, in which the convective term is approximated by the upwind difference derivative. The order of approximation of the convective term is improved due to a correction of the defect.
Key words:
singularly perturbed elliptic problem, grid approximation, domain decomposition method, special difference grids.
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Computational Mathematics and Mathematical Physics, 2005, 45:7, 1155–1171
Bibliographic databases:
UDC:
519.632.4 Received: 11.03.2003 Revised: 01.02.2005
Citation:
G. I. Shishkin, “Grid approximation of the domain and solution decomposition method with improved convergence rate for singularly perturbed elliptic equations in domains with characteristic boundaries”, Zh. Vychisl. Mat. Mat. Fiz., 45:7 (2005), 1196–1212; Comput. Math. Math. Phys., 45:7 (2005), 1155–1171
Citation in format AMSBIB
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\by G.~I.~Shishkin
\paper Grid approximation of the domain and solution decomposition method with improved convergence rate for singularly perturbed elliptic equations in domains with characteristic boundaries
\jour Zh. Vychisl. Mat. Mat. Fiz.
\yr 2005
\vol 45
\issue 7
\pages 11961212
\mathnet{http://mi.mathnet.ru/zvmmf625}
\mathscinet{http://www.ams.org/mathscinetgetitem?mr=2188412}
\zmath{https://zbmath.org/?q=an:1164.35313}
\transl
\jour Comput. Math. Math. Phys.
\yr 2005
\vol 45
\issue 7
\pages 11551171
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Kopteva N., O'Riordan E., “Shishkin meshes in the numerical solution of singularly perturbed differential equations”, Int. J. Numer. Anal. Model., 7:3 (2010), 393–415

U. H. Zhemuhov, “Uniform grid approximation of nonsmooth solutions with improved convergence for a singularly perturbed convectiondiffusion equation with characteristic layers”, Comput. Math. Math. Phys., 52:9 (2012), 1239–1259

Makarov S.S., Isaeva A.V., Grachev E.A., Serdobolskaya M.L., “Uskorenie vychislenii pri reshenii neodnorodnogo uravneniya diffuzii s pomoschyu perenormirovochnogo metoda”, Vychislitelnye metody i programmirovanie: novye vychislitelnye tekhnologii, 13:1 (2012), 239–246

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