Non-Polynomial Interpolation of Functions with Large Gradients and Its Application

Abstract

Interpolation of a function of one variable with large gradients in the boundary layer region is studied. The problem is that the use of classical polynomial interpolation formulas on a uniform mesh to functions with large gradients can lead to errors of the order of $$O(1)$$ , despite a small mesh size. An interpolation formula based on fitting to the component that defines the boundary-layer growth of the function is investigated. An error estimate, which depends on the number of interpolation nodes and is uniform over the boundary layer component and its derivatives, is obtained. It is shown how the interpolation formula derived can be used to construct formulas for numerical differentiation and integration and in the two-dimensional case. The corresponding error estimates are obtained.