The e-uniform convergence of the discrete derivatives for singularly perturbed problems

The derivatives of the solution of singularly perturbed differential equations become unbounded as the singular perturbation parameter e tends to zero. Therefore to approximate such derivatives, it is required to scale the derivatives in such a way that they are of order one for all values of the perturbation parameter. In practice, derivatives are related to the flux or drag and, hence, it is desirable to have e-uniform approximations to the scaled derivatives. In this paper, singularly perturbed convection--diffusion problems are considered. The use of standard scaled discrete derivatives to approximate the scaled continuous derivatives of the solution of singularly perturbed problems is examined. Standard scaled discrete derivatives generated from exact numerical methods on a uniform mesh are shown to be not e-uniformly convergent. On the other hand, standard scaled discrete derivatives computed from a numerical method based on an appropriately fitted piecewise--uniform mesh are shown to be e-uniformly convergent. Numerical results are presented and discussed to illustrate the significance of these theoretical results.