Approximation of systems of singularly perturbed elliptic reaction-diffusion equations with two parameters

G. I. Shishkin

Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 47 (2007) no. 5, pp. 835-866

Résumé

In a rectangle, the Dirichlet problem for a system of two singularly perturbed elliptic reaction-diffusion equations is considered. The higher order derivatives of the $i$th equation are multiplied by the perturbation parameter $\varepsilon_i^2$ ($i=1,2$). The parameters $\varepsilon_i$ take arbitrary values in the half-open interval $(0,1]$. When the vector parameter $\boldsymbol\varepsilon=(\varepsilon_1, \varepsilon_2)$ vanishes, the system of elliptic equations degenerates into a system of algebraic equations. When the components $\varepsilon_1$ and (or) $\varepsilon_2$ tend to zero, a double boundary layer with the characteristic width $\varepsilon_1$ and $\varepsilon_2$ appears in the vicinity of the boundary. Using the grid refinement technique and the classical finite difference approximations of the boundary value problem, special difference schemes that converge $\boldsymbol\varepsilon$-uniformly at the rate of $O(N^{-2}\ln^2N)$ are constructed, where $N=\min_sN_s$, $N_s+1$ is the number of mesh points on the axis $x_s$.