
This article is cited in 3 scientific papers (total in 3 papers)
Grid approximation in a half plane for singularly perturbed elliptic equations with convective terms that grow at infinity
G. I. Shishkin^{} ^{} Institute of Mathematics and Mechanics, Ural Division, Russian Academy of Sciences, ul. S. Kovalevskoi 16, Yekaterinburg, 620219, Russia
Abstract:
In the right half plane, the first boundaryvalue problem for singularly perturbed (with a perturbation parameter $\varepsilon\in(0,1]$) elliptic equations with convection terms is considered. The horizontal component of the convective flow is assumed to be directed towards the boundary and infinitely (linearly) increasing as $x_1\to\infty$. Two problems are examined: the problem with an explicitly observed reaction (the coefficient of the unknown function to be found is bounded away from zero, in which case the source is bounded in the domain) and the problem with a source of decreasing intensity (the source function decreases by a power law as $x_1\to\infty$, and the coefficient of the function to be found may be equal to zero in this case). For such problems, $\varepsilon$uniformly convergent formal and constructive finite difference schemes are constructed on grids with an infinite and finite, respectively, number of nodes. Monotone difference approximations of differential equations on piecewise uniform grids that condense in the boundary layer are used in the scheme construction. The resulting constructive schemes are convergent on given bounded subdomains. In the case of the problem with an explicitly observed reaction, these subdomains are chosen in the $\rho$ neighborhood of the domain boundary, where $\rho=o(N_1^{[0]})$ and $N_1^{[0]}$ is the number of nodes in the mesh in $x_1$ used in the constructive scheme. For problems with a source of decreasing intensity, no restrictions are imposed on the choice of those subdomains. When designing constructive schemes, we use the boundedness of the domain of essential dependence for both the solutions of the boundaryvalue problem and the formal difference schemes considered on the given bounded subdomains.
Key words:
Elliptic convectiondiffusion equations, singularly perturbed boundaryvalue problems, finite difference approximations, constructive difference schemes, unbounded domains.
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Computational Mathematics and Mathematical Physics, 2005, 45:2, 285–301
Bibliographic databases:
UDC:
519.632.4 Received: 11.03.2003 Revised: 25.08.2004
Citation:
G. I. Shishkin, “Grid approximation in a half plane for singularly perturbed elliptic equations with convective terms that grow at infinity”, Zh. Vychisl. Mat. Mat. Fiz., 45:2 (2005), 298–314; Comput. Math. Math. Phys., 45:2 (2005), 285–301
Citation in format AMSBIB
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This publication is cited in the following articles:

G. I. Shishkin, “Grid approximation of singularly perturbed parabolic reactiondiffusion equations on large domains with respect to the space and time variables”, Comput. Math. Math. Phys., 46:11 (2006), 1953–1971

G. I. Shishkin, “Approximation of singularly perturbed parabolic equations in unbounded domains subject to piecewise smooth boundary conditions in the case of solutions that grow at infinity”, Comput. Math. Math. Phys., 49:10 (2009), 1748–1764

Shishkin G.I., “Constructive and formal difference schemes for singularly perturbed parabolic equations in unbounded domains in the case of solutions growing at infinity”, Russian Journal of Numerical Analysis and Mathematical Modelling, 24:6 (2009), 591–617

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