Fast non-polynomial interpolation and integration for functions with logarithmic singularities

TL;DR

This paper introduces a rapid non-polynomial interpolation method tailored for functions with logarithmic singularities, utilizing the discrete cosine transform, and develops a new quadrature for such integrals, validated by numerical examples.

Contribution

It presents a novel fast non-polynomial interpolation technique and a corresponding quadrature for logarithmic singularities, with comprehensive error analysis and validation.

Findings

Interpolation and integration errors are minimized.

The method significantly improves computational efficiency.

Numerical examples confirm the accuracy and speed of the approach.

Abstract

A fast non-polynomial interpolation is proposed in this paper for functions with logarithmic singularities. It can be executed fast with the discrete cosine transform. Based on this interpolation, a new quadrature is proposed for a kind of logarithmically singular integrals. The interpolation and integration errors are also analyzed. Numerical examples of the interpolation and integration are shown to validate the efficiency of the proposed new interpolation and the new quadrature.