Necessary conditions for $\varepsilon$-uniform convergence of finite difference schemes for parabolic equations with moving boundary layers

G. I. Shishkin

Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 47 (2007) no. 10, pp. 1706-1726

Résumé

A grid approximation of the boundary value problem for a singularly perturbed parabolic reaction-diffusion equation is considered in a domain with the boundaries moving along the axis $x$ in the positive direction. For small values of the parameter $\varepsilon$ (this is the coefficient of the higher order derivatives of the equation, $\varepsilon\in(0,1]$), a moving boundary layer appears in a neighborhood of the left lateral boundary $S_1^L$. In the case of stationary boundary layers, the classical finite difference schemes on piece-wise uniform grids condensing in the layers converge $\varepsilon$-uniformly at a rate of $O(N^{-1}\ln N+N_0)$, where $N$ and $N_0$ define the number of mesh points in $x$ and $t$. For the problem examined in this paper, the classical finite difference schemes based on uniform grids converge only under the condition $N^{-1}+N_0^{-1}\ll\varepsilon$. It turns out that, in the class of difference schemes on rectangular grids that are condensed in a neighborhood of $S_1^L$ with respect to $x$ and $t$, the convergence under the condition $N^{-1}+N_0^{-1}\le\varepsilon^{1/2}$ cannot be achieved. Examination of widths that are similar to Kolmogorov's widths makes it possible to establish necessary and sufficient conditions for the $\varepsilon$-uniform convergence of approximations of the solution to the boundary value problem. These conditions are used to design a scheme that converges $\varepsilon$-uniformly at a rate of $O(N^{-1}\ln N+N_0)$.