This paper deals with convergence properties of the simple upwind difference scheme and a Galerkin finite element method (FEM) on generalized Shishkin grids. Conditions on the mesh-characterizing function that are sufficient for the convergence of the method, uniformly with respect to the perturbation parameter are presented. The authors verify experimentally the theoretical results for the Galerkin FEM and also test the performance of the simple upwind scheme on various meshes when applied to the test second-order layer problem for ordinary differential equations with homogeneous Dirichlet boundary conditions.