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Mathematical Problems in Engineering
Volume 2014 (2014), Article ID 490156, 10 pages
Research Article

A Smoothing Process of Multicolor Relaxation for Solving Partial Differential Equation by Multigrid Method

1Department of Engineering Mechanics, Kunming University of Science and Technology, Kunming, Yunnan 650500, China
2School of Mathematics and Computer, Dali University, Dali, Yunnan 671003, China

Received 24 June 2014; Accepted 26 August 2014; Published 25 September 2014

Academic Editor: Kim M. Liew

Copyright © 2014 Xingwen Zhu and Lixiang Zhang. 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.


This paper is concerned with a novel methodology of smoothing analysis process of multicolor point relaxation by multigrid method for solving elliptically partial differential equations (PDEs). The objective was firstly focused on the two-color relaxation technique on the local Fourier analysis (LFA) and then generalized to the multicolor problem. As a key starting point of the problems under consideration, the mathematical constitutions among Fourier modes with various frequencies were constructed as a base to expand two-color to multicolor smoothing analyses. Two different invariant subspaces based on the 2h-harmonics for the two-color relaxation with two and four Fourier modes were constructed and successfully used in smoothing analysis process of Poisson’s equation for the two-color point Jacobi relaxation. Finally, the two-color smoothing analysis was generalized to the multicolor smoothing analysis problems by multigrid method based on the invariant subspaces constructed.