Conditioning of a difference scheme of the solution decomposition method for a singularly perturbed convection-diffusion equation

Conditioning of a difference scheme of the solution decomposition method is studied for a Dirichlet problem for a singularly perturbed ordinary differential convection-diffusion equation. In this scheme, we apply a decomposition of the discrete solution into the regular and singular components, which are solutions of discrete subproblems, i.e., classical difference approximations considered on uniform grids. The scheme converges $\varepsilon$-uniformly in the maximum norm at the rate $\mathcal O(N^{-1}\ln N)$; $\varepsilon$ is a perturbation parameter multiplying the high-order derivative in the equation, $\varepsilon\in(0,1]$, and $N+1$ is the number of nodes in the grids used. It is shown that the solution decomposition scheme, unlike the standard scheme on uniform grid, is $\varepsilon$-uniformly well conditioned and stable to perturbations in the data of the discrete problem; the conditioning number of the scheme is a value of order $\mathcal O(\delta^{-2}\ln\delta^{-1})$, where $\delta$ is the accuracy of the discrete solution.