Two-Dimensional Interpolation of Functions by Cubic Splines in the Presence of Boundary Layers

We study interpolation of a function of two variables with large gradients in regions of a boundary layer under the assumption that the Shishkin decomposition into the sum of regular and boundary layer components is valid for the interpolated function. We generalize the one-dimensional cubic splines, studied earlier on the Shishkin and Bakhvalov grids, to the two-dimensional case. We obtain error estimates for a two-dimensional spline interpolation, uniform in a small parameter.