Superconvergence Points For The Spectral Interpolation Of Riesz Fractional Derivatives
For researchers in fractional calculus and numerical methods, this work provides a way to significantly improve accuracy of spectral approximations for Riesz derivatives, though it is an incremental extension of known superconvergence concepts.
This paper identifies superconvergence points for spectral interpolation of Riesz fractional derivatives, achieving convergence rate gains of O(N^{-2}) or better at these points compared to global rates, with theoretical analysis and validation on fractional differential equations.
In this paper, superconvergence points are located for the approximation of the Riesz derivative of order $α$ using classical Lobatto-type polynomials when $α\in (0,1)$ 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 $α\in (0,1)$ 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.