Uniform in a small parameter convergence of Samarskii's monotone scheme and its modification for the convection-diffusion equation with a concentrated source (Q5954354)

The first boundary value problem is considered on a segment for a one-dimensional convection-diffusion equation with a concentrated source. The problem is solved in the class of continuous and piecewise smooth functions. A modified monotone scheme is analyzed on nonuniform grids. This scheme is shown to be uniformly convergent with errors estimated as \(O(N^{-2}\ln{^2}N)\) and \(O(N^{-2})\), respectively.