Lagrange interpolation and the Newton–Cotes formulas on a Bakhvalov mesh in the presence of a boundary layer

Application of a Lagrange polynomial on a Bakhvalov mesh for the interpolation of a function with large gradients in an exponential boundary layer is studied. The problem is that the use of a Lagrange polynomial on a uniform mesh for interpolation of such a function can lead to errors of order $O(1)$, despite the smallness of the mesh size. The Bakhvalov mesh is widely used for the numerical solution of singularly perturbed problems, and the analysis of interpolation formulas on such a mesh is of interest. Estimates of the error of interpolation by a Lagrange polynomial with an arbitrary number of interpolation nodes on a Bakhvalov mesh are obtained. The result is used to estimate the error of the Newton–Cotes formulas on a Bakhvalov mesh. The results of numerical experiments are presented.