Approaches to constructing two-dimensional interpolation formulas in the presence of boundary layers

Abstract

An overview of approaches to the construction of two-dimensional interpolation formulas for a function of two variables with large gradients in the boundary layer regions is given. The problem is that the use of polynomial interpolation formulas on a uniform grid can lead to errors of the order of O(1). It is shown that the use of polynomial interpolation formulas on the Shishkin and Bakhvalov grids leads to the fact that the error becomes uniform in a small parameter. More accurate results are obtained by using the Bakhvalov grid. It is shown that on a uniform grid it is possible to successfully apply interpolation formulas that are exact on the singular components responsible for large function gradients in boundary layers. Numerical results are given.