International Journal of Mathematics and Mathematical Sciences
Volume 23 (2000), Issue 5, Pages 343-359
doi:10.1155/S0161171200001940
Superconvergence of a finite element method for linear integro-differential problems
1Department of Mathematics, Kaist, Taejon 305-701, Korea
2Department of Mathematics, Shandong Normal University, Shandong, Jinan 250014, China
Received 10 September 1998
Copyright © 2000 Do Y. Kwak et al. 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.
Abstract
We introduce a new way of approximating initial condition to the
semidiscrete finite element method for integro-differential
equations using any degree of elements. We obtain several
superconvergence results for the error between the approximate
solution and the Ritz-Volterra projection of the exact solution.
For k>1, we obtain first order gain in Lp(2≤p≤∞) norm, second order in W1,p(2≤p≤∞) norm and almost
second order in W1,∞ norm. For k=1, we obtain first order gain in W1,p(2≤p≤∞) norms. Further,
applying interpolated postprocessing technique to the approximate
solution, we get one order global superconvergence between the
exact solution and the interpolation of the approximate solution
in the Lp and W1,p(2≤p≤∞).