Grid approximation in a half plane for singularly perturbed elliptic equations with convective terms that grow at infinity

G. I. Shishkin

Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 45 (2005) no. 2, pp. 298-314

Résumé

In the right half plane, the first boundary-value 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 boundary-value problem and the formal difference schemes considered on the given bounded subdomains.

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