Studies on Barlow points, Gauss points and Superconvergent points in 1D with Lagrangian and Hermitian finite element basis

In this work, we consider the superconvergence property of the finite element derivative for Lagrange's and Hermite's Family elements in the one dimensional interpolation problem. We also compare the Barlow points, Gauss points and Superconvergence points in the sense of Taylor's Series, confirming that they are not the same as believed before. We prove a not evident and new superconvergence property of Hermite's basis as well which shows that the centroid is not only a superconvergent for u'h but an O(h5) accuracy point.