Grid approximation of singularly perturbed equations, degenerated on the boundary. The case of sharply changing coefficients in the neighbourhood of the boundary layer

G. I. Shishkin

Matematičeskoe modelirovanie, Tome 6 (1994) no. 5, pp. 105-121.

Résumé

On rectangular domain $G$, $G=(0,d_1]\times(0,d_2]$, tne Dirichlet problem for singularly perturbed equation of parabolic type $\{\varepsilon\partial_1^2-b(x_1)\partial/\partial x_2\}u(x)=f(x)$, where $b(x_1)=\min[(\sigma^{-1}x_1)^\alpha,1]$ is considered. The partial differential equation is degenerated into the second order ordinary differential equation when $x_1=0$; $x_2$ a time variable, the parameters $\varepsilon$, $\sigma$ can get any value on intervals $(0,1]$ and $[0,d_1/2]$ respectively, $\alpha\in(0,M]$, $M>1$. When $\varepsilon=0$ reduced first order equation is degenerated on the boundary domain for $x_1=0$. The difference scheme (on the grids condensing in the boundary and interior layers) is constructed which converges uniformly with respect to the parameters $\varepsilon$ and $\sigma$. Also grid approximations of the boundary value problems for elliptic equation $\{\varepsilon\Delta-b(x_1)\partial/\partial x_2\}u(x)=f(x)$ are considered. The problems of investigated type appear, for example, when the diffusion processes in moving medium are modelled.