FORMULAS FOR NUMERICAL DIFFERENTIATION ON A UNIFORM GRID IN THE PRESENCE OF A BOUNDARY LAYER

The problem of numerical differentiation of functions with large gradients is considered. It is assumed that for the original function of one variable the decomposition is valid as the sum of a regular component with bounded derivatives up to a certain order and a boundary layer component having large gradients and known with an accuracy of up to a factor. Such a decomposition, in particular, is valid for solution of a singularly perturbed boundary value problem. The topic of the study is relevant, since the application of classical polynomial formulas of numerical differentiation to functions with large gradients can lead to significant errors. The error of the formulas of numerical differentiation, according to the construction of exact ones on the boundary layer component of the original function, is estimated. The results of numerical experiments are presented, consistent with the obtained error estimates.