Improved difference scheme of the solution decomposition method for a singularly perturbed reaction-diffusion equation

A Dirichlet problem is considered for a singularly perturbed ordinary differential reaction-diffusion equation. For this problem, a new approach is developed in order to construct difference schemes convergent uniformly with respect to the perturbation parameter $\varepsilon$, $\varepsilon\in(0,1]$. The approach is based on the decomposition of a discrete solution into regular and singular components, which are solutions of discrete subproblems on uniform grids. Using the asymptotic construction technique, a difference scheme of the solution decomposition method is constructed that converges $\varepsilon$-uniformly in the maximum norm at the rate $\mathcal O(N^{-2}\ln^{-2}N)$, where $N+1$ is the number of nodes in the grids used; for fixed values of the parameter $\varepsilon$, the scheme converges at the rate $\mathcal O(N^{-2})$. Using the Richardson technique, an improved scheme of the solution decomposition method is constructed, which converges $\varepsilon$-uniformly in the maximum norm at the rate $\mathcal O(N^{-4 }\ln^{-4}N)$.