Journal of Inequalities and Applications
Volume 2010 (2010), Article ID 895187, 17 pages
doi:10.1155/2010/895187
Research Article

L2-Error Estimates of the Extrapolated Crank-Nicolson Discontinuous Galerkin Approximations for Nonlinear Sobolev Equations

Mi Ray Ohm,1 Hyun Young Lee,2 and Jun Yong Shin3

1Division of Information Systems Engineering, Dongseo University, 617-716 Busan, South Korea
2Department of Mathematics, Kyungsung University, 608-736 Busan, South Korea
3Division of Mathematical Sciences, Pukyong National University, 608-737 Busan, South Korea

Received 9 November 2009; Accepted 23 January 2010

Academic Editor: Jong Kim

Copyright © 2010 Mi Ray Ohm 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 analyze discontinuous Galerkin methods with penalty terms, namely, symmetric interior penalty Galerkin methods, to solve nonlinear Sobolev equations. We construct finite element spaces on which we develop fully discrete approximations using extrapolated Crank-Nicolson method. We adopt an appropriate elliptic-type projection, which leads to optimal ℓ∞(L2) error estimates of discontinuous Galerkin approximations in both spatial direction and temporal direction.