Application of rational interpolations for solving boundary value problems with singularities

In this paper, we develop, implement and test a new method for solving singularly perturbed boundary value problems for non-linear partial differential equations of the second order, which are posed in a rectangular domain. In order to approximate a solution, the tensor products of rational functions are used in the method. These functions are obtained from barycentric interpolation polynomials with Chebyshev nodes by means of a special change of variable. The purpose of this change of variable is to adapt the locations of interpolation nodes to singularities of the desired function, which leads to concentration of them in the neighborhood of large gradients of the solution. To approximate the non-linear equations, a combination of iterative and collocation methods is used. This allows to pass to the matrix Sylvester equation at each iteration and to reduce considerably the run time of algorithm. High computational performance of the method is demonstrated on the example of test boundary value problem in square domain with a known solution, which has a peak in the centre of the domain. Such singular behaviour is related with the presence of pole of the desired function in complex plain.