In this paper, a new Legendre orthonormal polynomial–based least–squares method(LSM) is provided for fractional integro-differential equations(FIDEs). We first divide the domain into $n$ cells, and in each cell we apply the least–squares algorithm to obtain an approximating solution. The stability and the optimal convergence order in $W_2^2$–norm error are proven. Numerical examples are studied to verify our theoretical discovery. Comparison with the sextic and eighth $C^{3}-$spline method illustrates that our algorithm can obtain a more accurate approximating solution.

In this paper, we analyze two classes of spectral volume (SV) methods for one-dimensional hyperbolic equations with degenerate variable coefficients. The two classes of SV methods are constructed by letting a piecewise $k$-th order ($k\ge 1$ is an arbitrary integer) polynomial to satisfy the conservation law in each {\it control volume}, which is obtained by refining spectral volumes (SV) of the underlying mesh with $k$ Gauss-Legendre points (LSV) or Radaus points (RSV) in each SV. The $L^2$-norm stability and optimal order convergence properties for both methods are rigorously proved for general non-uniform meshes. Surprisingly, we discover some very interesting superconvergence phenomena : at some special points, the SV flux function approximates the exact flux with $(k+2)$-th order and the SV solution itself approximates the exact solution with $(k+\frac32)$-th order; some superconvergence behaviors for element averages errors have been also discovered. Moreover, these superconvergence phenomena are rigorously proved by using the so-called {\it correction function} method. Our theoretical findings are verified by several numerical experiments.