The Lagrange interpolation and the Newton–Cotes formulas for functions with a boundary layer component on piecewise-uniform meshes

The interpolation problem of a one-variable function, which can be considered as a solution of a boundary value problem for an equation with a small parameter $\varepsilon$ with a higher derivative is investigated. The application of the Lagrange interpolation for such a function on a uniform grid can result in serious errors. In the case of the Shishkin mesh, $\varepsilon$-uniform error estimates of the Lagrange interpolation are obtained. The Shishkin mesh is modified to increase the interpolation accuracy. The $\varepsilon$-uniform error estimates of the Newton–Cotes formulas on such meshes are obtained. Numerical experiments have been carried out. The results obtained confirm the theoretical estimates.