A linear barycentric rational interpolant on starlike domains

TL;DR

This paper develops a new two-dimensional rational interpolant on starlike domains using conformally shifted Chebyshev and equispaced nodes, achieving exponential convergence for analytic functions and high accuracy with fewer nodes.

Contribution

It introduces a novel tensor-product interpolant combining rational barycentric schemes with shifted nodes, proven to converge exponentially on analytic functions in two dimensions.

Findings

Exponential convergence for analytic functions on starlike domains.

Fewer nodes needed for high accuracy due to shifted nodes.

Numerical results confirm improved efficiency and precision.

Abstract

When an approximant is accurate on the interval, it is only natural to try to extend it to several-dimensional domains. In the present article, we make use of the fact that linear rational barycentric interpolants converge rapidly toward analytic and several times differentiable functions to interpolate on two-dimensional starlike domains parametrized in polar coordinates. In radial direction, we engage interpolants at conformally shifted Chebyshev nodes, which converge exponentially toward analytic functions. In circular direction, we deploy linear rational trigonometric barycentric interpolants, which converge similarly rapidly for periodic functions, but now for conformally shifted equispaced nodes. We introduce a variant of a tensor-product interpolant of the above two schemes and prove that it converges exponentially for two-dimensional analytic functions -- up to a logarithmic factor -- and with an order limited only by the order of differentiability for real functions (provided that the boundary enjoys the same order of differentiability). Numerical examples confirm that the shifts permit to reach a much higher accuracy with significantly fewer nodes, a property which is especially important in several dimensions.