On the uniform convergence of parabolic spline interpolation on the class of functions with large gradients in the boundary layer

Abstract

The problem of parabolic spline interpolation, according to Subbotin, of functions with large gradients in the boundary layer is considered. In the case of a uniform grid it is proved and in the case of the Shishkin mesh it is experimentally shown that with a parabolic spline interpolation of functions with large gradients in the exponential boundary layer, the error can unrestrictedly increase when the small parameter tends to zero and the number of grid nodes is fixed. An approximation process using parabolic splines with defect 1 is proposed; the error estimates are found to be uniform in the small parameter.