Grid approximations with an improved rate of convergence for singularly perturbed elliptic equations in domains with characteristic boundaries

On a rectangle, we consider the Dirichlet problem for singularly perturbed elliptic equations with convective terms in the case of characteristics of the reduced equations which are parallel to the sides. For such convection-diffusion problems the uniform (with respect to the perturbation parameter $\varepsilon$) convergence rate of the well-known special schemes on piecewise uniform meshes is of order not higher than one (in the uniform $L_{\infty}$-norm). For the above problem, based on asymptotic expansions of the solutions, we construct schemes that converge $\varepsilon$-uniformly with the rate $\mathscr O(N^{-2}\ln^2N)$, where $N$ defines the number of mesh points with respect to each variable. For not too small values of the parameter we apply classical finite difference approximations on piecewise uniform meshes condensing in boundary layers; for small values of the parameter we use approximations of auxiliary problems, which describe the main terms of asymptotic representation of the solution in a neighborhood of the boundary layer and outside of it. Note that the computation of solutions of the constructed difference scheme is simplified for sufficiently small values of the parameter $\varepsilon$.