Superconvergence Points for the Spectral Interpolation of Riesz Fractional Derivatives∗

In this paper, superconvergence points are located for the approximation of the Riesz derivative of order α using classical Lobatto-type polynomials when α ∈ (0, 1) and generalized Jacobi functions (GJF) for arbitrary α > 0, respectively. For the former, superconvergence points are zeros of the Riesz fractional derivative of the leading term in the truncated Legendre-Lobatto expansion. It is observed that the convergence rate for different α at the superconvergence points is at least O(N−2) better than the optimal global convergence rate. Furthermore, the interpolation is generalized to the Riesz derivative of order α > 1 with the help of GJF, which deal well with the singularities. The well-posedness, convergence and superconvergence properties are theoretically analyzed. The gain of the convergence rate at the superconvergence points is analyzed to be O(N−(α+3)/2) for α ∈ (0, 1) and O(N−2) for α > 1. Finally, we apply our findings in solving model FDEs and observe that the convergence rates are indeed much better at the predicted superconvergence points.