Two-dimensional interpolation of functions with large gradients in boundary layers

The question of interpolation of a function of two variables with large gradients in the regions of the boundary layer is considered. The interpolated function has a representation as a sum of regular and boundary layer components. Such the representation is valid due to the Shishkin decomposition for the solution of the singularly perturbed problem. The development of interpolation formulas for such functions is relevant, since in the case of a uniform grid the error can be of the order of $O(1).$ In the rectangular domain a Bakhvalov mesh is applied, which condenses in the boundary layers. The initial domain is divided into rectangular cells. In each such cell, two-dimensional interpolation based on the Lagrange polynomial is applied. The interpolation formula contains $k$ interpolation nodes in each direction. For each cell, an error estimate is obtained taking into account uniformity in a small parameter. An estimate of the stability of the interpolation formula is obtained on a two-dimensional grid from the class of Bakhvalov grids. The results of numerical experiments are consistent with the obtained error estimates. The study of the interpolation formula is necessary to continue the solution of the difference scheme from the grid nodes to the entire original domain.