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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 391918, 11 pages
http://dx.doi.org/10.1155/2012/391918
Research Article

A Two-Grid Method for Finite Element Solutions of Nonlinear Parabolic Equations

1Department of Mathematics and Information Science, Yantai University, Yantai 264005, China
2Schools of Mathematics and Quantitative Economics, Shandong University of Finance, Jinan 250014, China

Received 5 July 2012; Accepted 24 August 2012

Academic Editor: Xinguang Zhang

Copyright © 2012 Chuanjun Chen and Wei Liu. 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

A two-grid method is presented and discussed for a finite element approximation to a nonlinear parabolic equation in two space dimensions. Piecewise linear trial functions are used. In this two-grid scheme, the full nonlinear problem is solved only on a coarse grid with grid size . The nonlinearities are expanded about the coarse grid solution on a fine gird of size , and the resulting linear system is solved on the fine grid. A priori error estimates are derived with the -norm which shows that the two-grid method achieves asymptotically optimal approximation as long as the mesh sizes satisfy . An example is also given to illustrate the theoretical results.