Analysis of polynomial interpolation of the function of two variables with large gradients in the parabolic boundary layers

Abstract

The problem of interpolation of the function of two variables with large gradients in the parabolic and exponential boundary layers is investigated. It is assumed that the function has large gradients near the boundaries of a rectangular domain. Such function corresponds to the solution of the convection-diffusion problem with dominant convection. It is known that the error of polynomial interpolation on uniform grid for such function can be of the order of O(1). We propose to use two-dimensional polynomial interpolation on the Shishkin mesh. The error estimate uniform with respect to the perturbation parameter is obtained. Numerical results are presented to validate the theoretical results.