On the theory of difference schemes for singularly perturbed equations

We consider a two-point boundary value problem for a linear singularly perturbed reaction-diffusion equation. For the numerical solution of this problem, we use a classical three-point difference scheme on an arbitrary nonuniform grid. We introduce the so-called weighted W 1,∞;ε2 h-norm equal to the sum of the negative W -1,∞ h-norm of the grid function and the L ∞ h-norm of its first difference multiplied by the small parameter ε 2. We prove a uniform (with respect to the small parameter) two-sided a priori estimate of this norm of the grid solution via the W -1,∞ h-norm of the right-hand side. The a priori estimate is obtained with the use of the Green function of the grid problem, for which we also obtain appropriate estimates in the corresponding anisotropic norms. We show that the grid solution ε-uniformly converges at the rate O(N -2) in the L ∞ h-norm, where N is the number of grid nodes, on a nonuniform grid condensing no worse than the Bakhvalov grid in neighborhoods of boundary layers and fairly arbitrary in other respects.