The author deals with the application of finite difference schemes applied to singulary perturbed boundary value problems. The numerical methods use five-point formulae inside the boundary layer and lower-order formulae elsewhere. The schemes are of consistency order 6 with a constant being independent of the step size \(\delta\) and the singular perturbation parameter \(\varepsilon\). Moreover, under the condition \(\delta^2\leq\varepsilon\) a stability inequality and, thus, convergence is shown. Finally, some numerical examples are presented to show the real convergence order for fixed pairs of \(\varepsilon\) and \(\delta\) in various situations.