Superconvergence points of fractional spectral interpolation

Xuan Zhao, Zhimin Zhang

Published 2015-03-24Version 1

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.