Interpolation of functions with the boundary layer components and its application in a two-grid method

As known, elliptic equation with regular boundary layers can be solved using difference scheme on an uniform mesh or on a mesh, dense in boundary layers. In both cases we have to solve a linear system of equations by iterations. We can reduce a number of iterations, if we preliminarily solve a problem on a coarse mesh. In this case we need to interpolate the mesh solution from a coarse mesh to a fine mesh. In a case of the uniform mesh we construct interpolations, fitted to the boundary layer components. We prove that in a case of Shishkin mesh the polynomial interpolation has the property of an uniform accuracy and may be used in a two-grid method.