A Dirichlet problem is considered for a singularly perturbed ordinary differential reaction-diffusion equation. For this problem, a new approach is developed to construct difference
schemes convergent uniformly with respect to a perturbation parameter ε for ε ∈ (0, 1], i.e.,
ε-uniformly. This approach is based on a decomposition of the discrete solution into the regular
and singular components which are solutions of discrete subproblems considered on uniform
grids. Using an asymptotic construction technique, a difference scheme of the solution decomposition
method is constructed that converges ε-uniformly in the maximum norm at the rate
O(N^{-2}ln^2N), where N + 1 is the number of nodes in the grids used; for fixed values of the
parameter ε, the scheme converges at the rate O(N^{-2}). For the constructed scheme, approximations
of the regular and singular components to the solution and their derivatives up to the
second order are studied. A modified scheme of the solution decomposition method is constructed
for which the regular component of the solution and its discrete derivatives converge $\varepsilon$-uniformly
in the maximum norm at the rate O(N^{-2}) for ε = o(ln^{-1} N).
