Journal of Inequalities and Applications
Volume 3 (1999), Issue 1, Pages 91-107

On the sharpness of a superconvergence estimate in connection with one-dimensional Galerkin methods

St. J. Goebbels

Lehrstuhl A für Mathematik, RWTH Aachen, Templergraben 55, Aachen D-52062, Germany

Received 5 November 1997; Revised 8 January 1998

Copyright © 1999 St. J. Goebbels. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


The present paper studies some aspects of approximation theory in the context of one-dimensional Galerkin methods. The phenomenon of superconvergence at the knots is well-known. Indeed, for smooth solutions the rate of convergence at these points is O(h2r) instead of O(hr+1), where r is the degree of the finite element space. In order to achieve a corresponding result for less smooth functions, we apply K-functional techniques to a Jackson-type inequality and estimate the relevant error by a modulus of continuity. Furthermore, this error estimate requires no additional assumptions on the solution, and it turns out that it is sharp in connection with general Lipschitz classes. The proof of the sharpness is based upon a quantitative extension of the uniform boundedness principle in connection with some ideas of Douglas and Dupont [Numer. Math. 22] (1974) 99–109. Here it is crucial to design a sequence of test functions such that a Jackson–Bernstein-type inequality and a resonance condition are satisfied simultaneously.