Grid approximation of a singularly perturbed elliptic equation with convective terms in the presence of various boundary layers

G. I. Shishkin

Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 45 (2005) no. 1, pp. 110-125

Résumé

The Dirichlet problem for a singularly perturbed elliptic convection-diffusion equation in a rectangle and in a vertical half-strip with a vector perturbation parameter $\varepsilon=(\varepsilon_1,\varepsilon_2)$ is considered. The higher derivatives of the equation and the first derivative with respect to the vertical coordinate include the parameters $\varepsilon_1$ and $\varepsilon_2$, respectively, which can take arbitrary values in the intervals $(0,1]$ and $[--1,1]$. For small values of $\varepsilon_1$, boundary layers appear in the neighborhood of various parts of the domain boundary. The type of these layers depends on the relation between $\varepsilon_1$ and $\varepsilon_2$: they can be regular, parabolic, or hyperbolic. Their characteristics also depend on the relation between $\varepsilon_1$ and $\varepsilon_2$. Using the special grid technique (these grids are condensing in the boundary layers), finite difference schemes are constructed that $\varepsilon$-uniformly converge in the maximum norm.