A conservative difference scheme for a singularly perturbed elliptic reaction-diffusion equation: approximation of solutions and derivatives

G. I. Shishkin ; L. P. Shishkina

Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 50 (2010) no. 4, pp. 665-678

Résumé

A boundary value problem for a singularly perturbed elliptic reaction-diffusion equation in a vertical strip is considered. The derivatives are written in divergent form. The derivatives in the differential equation are multiplied by a perturbation parameter $\varepsilon^2$, where $\varepsilon$ takes arbitrary values in the interval $(0, 1]$. As $\varepsilon\to0$, a boundary layer appears in the solution of this problem. Using the integrointerpolational method and the condensing grid technique, conservative finite difference schemes on flux grids are constructed that converge $\varepsilon$-uniformly at a rate of $O(N_1^{-2}\ln^2N_1+N_2^{-2})$, where $N_1+1$ and $N_2+1$ are the number of mesh points on the $x_1$-axis and the minimal number of mesh points on a unit interval of the $x_2$-axis respectively. The normalized difference derivatives $\varepsilon^k(\partial^k/\partial x_1^k)u(x)$ ($k = 1$, $2$), which are $\varepsilon$-uniformly bounded and approximate the normalized derivatives in the direction across the boundary layer, and the derivatives along the boundary layer $(\partial^k/\partial x_2^k)u(x)$ ($k = 1$, $2$) converge $\varepsilon$-uniformly at the same rate.