Computational methods and algorithms

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

Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences

Abstract: 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.

UDC: 519.633

Received: 11.11.1992

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