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

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 ith equation are multiplied by the perturbation parameter ? _i ² (i = 1, 2). The parameters ?_i take arbitrary values in the half-open interval (0, 1]. When the vector parameter ? = (?₁, ?₂) vanishes, the system of elliptic equations degenerates into a system of algebraic equations. When the components ?₁ and (or) ?₂ tend to zero, a double boundary layer with the characteristic width ?₁ and ?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 ?-uniformly at the rate of O(N ^?2ln² N) are constructed, where N = min N _s, N _s + 1 is the number of mesh points on the axis x _s.