Superconvergence of spectral collocation and $p$-version methods in one dimensional problems

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Superconvergence phenomenon of the Legendre spectral collocation method and the p-version finite element method is discussed under the one dimensional setting. For a class of functions that satisfy a regularity condition (M): ∥u (k) ∥L∞ ≤ cM k on a bounded domain, it is demonstrated, both theoretically and numerically, that the optimal convergent rate is supergeometric. Furthermore, at proper Gaussian points or Lobatto points, the rate of convergence may gain one or two orders of the polynomial degree.