An application of the cubic spline on Shishkin mesh for the approximation of a function and its derivatives in the presence of a boundary layer

The problem of the approximate calculation of the derivatives of functions having large gradients in the region of an exponential boundary layer is considered. The problem is that the application of the classical formulas of the numerical differentiation on the uniform grids to functions with large gradients leads to significant errors. It is proposed to apply a cubic spline interpolation on a Shishkin grid that condensed in the boundary layer. It is proposed to approximate the derivatives on the basis of the spline differentiation. The error of such approximation is estimated taking into account the uniformity of the estimate with respect to the small parameter.