Approximation of a system of singularly perturbed reaction-diffusion parabolic equations in a rectangle

The Dirichlet problem for a system of singularly perturbed reaction-diffusion parabolic equations in a rectangle is considered. The higher order derivatives of the equations are multiplied by a perturbation parameter $\varepsilon^2$, where $\varepsilon$ takes arbitrary values in the interval (0, 1]. When $\varepsilon$ vanishes, the system of parabolic equations degenerates into a system of ordinary differential equations with respect to $t$. When $\varepsilon$ tends to zero, a parabolic boundary layer with a characteristic width $\varepsilon$ appears in a neighborhood of the boundary. Using the condensing grid technique and the classical finite difference approximations of the boundary value problem, a special difference scheme is constructed that converges $\varepsilon$-uniformly at a rate of $O(N^{-2}\ln^2N+N_0^{-1})$, where $N=\min_s N_s$, $N_s+1$ and $N_s+1$ are the numbers of mesh points on the axes $x_s$ and $t$, respectively.