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Mathematical Problems in Engineering
Volume 2015, Article ID 572198, 7 pages
Research Article

Smoothing Analysis of Distributive Red-Black Jacobi Relaxation for Solving 2D Stokes Flow 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 15 September 2014; Revised 7 March 2015; Accepted 8 March 2015

Academic Editor: Vassilios C. Loukopoulos

Copyright © 2015 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.


Smoothing analysis process of distributive red-black Jacobi relaxation in multigrid method for solving 2D Stokes flow is mainly investigated on the nonstaggered grid by using local Fourier analysis (LFA). For multigrid relaxation, the nonstaggered discretizing scheme of Stokes flow is generally stabilized by adding an artificial pressure term. Therefore, an important problem is how to determine the zone of parameter in adding artificial pressure term in order to make stabilization of the algorithm for multigrid relaxation. To end that, a distributive red-black Jacobi relaxation technique for the 2D Stokes flow is established. According to the 2-harmonics invariant subspaces in LFA, the Fourier representation of the distributive red-black Jacobi relaxation for discretizing Stokes flow is given by the form of square matrix, whose eigenvalues are meanwhile analytically computed. Based on optimal one-stage relaxation, a mathematical relation of the parameter in artificial pressure term between the optimal relaxation parameter and related smoothing factor is well yielded. The analysis results show that the numerical schemes for solving 2D Stokes flow by multigrid method on the distributive red-black Jacobi relaxation have a specified convergence parameter zone of the added artificial pressure term.