<p>In this paper a singularly perturbed convection-diffusion problem with non-smooth data is studied. A second-order three-point difference method on an adjusted arbitrary mesh is utilized. A posteriori error analysis is investigated for the three-point difference scheme. The generation of a posteriori adaptive mesh and approximation solution is achieved by solving an equidistribution problem based on a posteriori error estimation and a mesh adjustment technique. Numerical experiments validate that the scheme is second-order uniformly convergent and is more accurate than the method on a posteriori mesh constructed in segments.</p>