Superconvergence points of fractional spectral interpolation

TL;DR

This paper studies superconvergence points in fractional spectral interpolation, identifying where derivatives or function values superconverge, using Legendre polynomials and Petrov-Galerkin methods with generalized Jacobi functions, supported by numerical verification.

Contribution

It introduces new superconvergence points for fractional derivatives and function values in spectral interpolation, unifying and extending previous results with novel methods.

Findings

Superconvergence points depend on fractional derivative order.

Legendre polynomial basis yields superconvergence points for derivatives.

Petrov-Galerkin method locates superconvergence points for function values and derivatives.

Abstract

We investigate superconvergence properties of the spectral interpolation involving fractional derivatives. Our interest in this superconvergence problem is, in fact, twofold: when interpolating function values, we identify the points at which fractional derivatives of the interpolant superconverge; when interpolating fractional derivatives, we locate those points where function values of the interpolant superconverge. For the former case, we apply various Legendre polynomials as basis functions and obtain the superconvergence points, which naturally unify the superconvergence points for the first order derivative presented in [Z. Zhang, SIAM J. Numer. Anal., 50 (2012), 2966-2985], depending on orders of fractional derivatives. While for the latter case, we utilize Petrov-Galerkin method based on generalized Jacobi functions (GJF) [S. Chen et al., arXiv: 1407. 8303v1] and locate the superconvergence points both for function values and fractional derivatives. Numerical examples are provided to verify the analysis of superconvergence points for each case.