Interpolation of a function of two variables with large gradients in boundary layers

Abstract

The paper is concerned with the interpolation of a function of two variables with large gradients in the boundary layers. An underlying function is assumed to be the sum of the regular componentwith derivatives bounded up to some order and two boundary layer components, the latter are known up to a multiplicative factor. Such a representation is typical for the solution of a singular perturbed elliptic problem. A two-dimensional interpolation formula exact on the boundary layer components is put forward. The formula has an arbitrary number of nodes in each direction. An error estimate is obtained which is uniform on the gradients of the underlying function in boundary layers. Results of numerical experiments are provided.