SOME SUPERCONVERGENCE FOR A GALERKIN METHOD BY AVERAGING GRADIENTS IN ONE DIMENSIONAL PROBLEMS.

We consider some superconvergence phenomena followed by the averaging gradients in a Galerkin method for two point boundary value problems using continuous piecewise polynomials. It is shown that several a posteriori methods based on the averaging procedures yield superconvergent approximations to the exact solution and its derivative with one order better rates of convergence than the optimal rates. The special emphasis of the paper is the fact that the superconvergence phenomena only occur in cases using odd degree polynomials. We describe the extension of the results to the parabolic problems in a single space variable.