A linear barycentric rational interpolant on starlike domains

Electronic Transactions on Numerical Analysis

article

Jean-Paul Berrut¹  Giacomo Elefante² 

[1] Department of Mathematics, UniversitÃl’ de Fribourg;Department of Mathematics “Tullio Levi-Civita”, Università di Padova

关键词: barycentric rational interpolation;    trigonometric interpolation;    Lebesgue constant;    conformal maps;    starlike domains;   

DOI  :  10.1553/etna_vol55s726

学科分类：数学（综合）

来源: Kent State University * Institute of Computational Mathematics

【 摘 要 】

When an approximant is accurate on an interval, it is only natural to try to extend it to multi-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 them on two-dimensional starlike domains parametrized in polar coordinates. In the radial direction, we engage interpolants at conformally shifted Chebyshev nodes, which converge exponentially for analytic functions. In the 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 one to reach a much higher accuracy with significantly fewer nodes, a property which is especially important in several dimensions.

【 授权许可】

Unknown