The article reviews the most important results in the field of numerical mathematics obtained by Professor V. V. Strygin’s disciples and the disciples of his disciples in 2000–2019 in the city of Samara. We present the results on spline interpolation of functions with a boundary layer, methods and algorithms for a posteriori adaptation of computational grids for singularly perturbed boundary value problems and the results on applied wavelet analysis based on spline wavelets.

The problem of cubic spline interpolation on the Bakhvalov mesh of functions with region of large gradients is considered. Asymptotically accurate error estimates O(N −4) are obtained for a class of functions with an exponential boundary layer in case 1/N ≤ ε, where N is number of nodes, ε is small parameter. In case ε ≤ 1/N we have experimentally shown that the error estimates of traditional spline interpolation are not uniform in a small parameter, and the error itself can increase indefinitely when the small parameter tends to zero at a fixed number of nodes N. A modified cubic spline is proposed for which uniform estimates of the order O(N −4) have been experimentally confirmed.

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