Formulas for Numerical Differentiation on a Uniform Mesh in the Presence of a Boundary Layer

Numerical differentiation of functions with large gradients is considered. It is assumed that the original function of one variable can be decomposed into the sum of a regular component with bounded derivatives up to a certain order and a boundary layer component, which has large gradients and is known up to a factor. In particular, this decomposition is relevant for solution of a singularly perturbed boundary value problem, since the application of classical polynomial formulas of numerical differentiation to functions with large gradients can lead to significant errors. The error of numerical differentiation formulas is estimated for constructed formulas exact on the boundary layer component of the original function. The results of numerical experiments, consistent with the error estimates obtained, are presented.