The epsilon-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 " 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 "-- 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 "--uniformly convergent. On the other hand, standard scaled discrete derivatives computed from a numerical method based on an appropriately fitted piecewise--uniform mesh ..

A Technique for Computing Realistic Values of the Error Parameters for the Numerical Solutions of Singular Perturbation Problems

. In this paper we describe an experimental technique to determine approximate values of the error parameters associated with a parameter--uniform numerical method for solving singularly perturbed convection--diffusion problems. We employ the technique to compute realistic values of these parameters for the numerical solutions generated by a monotone parameter--uniform numerical method applied to an elliptic boundary value problem with different types of boundary layers such as regular, parabolic and corner layers. Such error parameters allow us effectively to evaluate actual error bounds for the numerical solutions and to determine the parameter-uniformity of new numerical methods and, therefore, their applicability in practice. 1 Introduction The numerical solution of a singularly perturbed problem and its actual maximum pointwise error depend on the singular perturbation parameter " and the number N of mesh points in each coordinate direction of the discrete problem. A standard cri..