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.
Gauss points; Barlow points; Superconvergence points; derivative; finite elements
