On node distributions for interpolation and spectral methods

TL;DR

This paper investigates a scaled Chebyshev node distribution, proving its optimality for interpolation of certain smooth functions and demonstrating that it improves accuracy in spectral differentiation over traditional Chebyshev-Gauss-Lobatto nodes.

Contribution

It introduces a scaled Chebyshev node distribution, proves its optimality for interpolation, and shows its effectiveness in spectral differentiation through numerical experiments.

Findings

Scaled Chebyshev nodes are optimal for interpolation in $C_M^{s+1}[-1,1]$.

Proposed nodes improve spectral differentiation accuracy.

Numerical results outperform Chebyshev-Gauss-Lobatto nodes.

Abstract

A scaled Chebyshev node distribution is studied in this paper. It is proved that the node distribution is optimal for interpolation in $C_M^{s+1}[-1,1]$, the set of $(s+1)$-time differentiable functions whose $(s+1)$-th derivatives are bounded by a constant $M>0$. Node distributions for computing spectral differentiation matrices are proposed and studied. Numerical experiments show that the proposed node distributions yield results with higher accuracy than the most commonly used Chebyshev-Gauss-Lobatto node distribution.