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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 867598, 14 pages
http://dx.doi.org/10.1155/2012/867598
Research Article

Convergence Analysis of the Preconditioned Group Splitting Methods in Boundary Value Problems

School of Mathematical Sciences, Universitiy Sains Malaysia, 11800 USM, Pulau Pinang, Malaysia

Received 17 May 2012; Accepted 12 July 2012

Academic Editor: Ravshan Ashurov

Copyright © 2012 Norhashidah Hj. Mohd Ali and Abdulkafi Mohammed Saeed. 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

The construction of a specific splitting-type preconditioner in block formulation applied to a class of group relaxation iterative methods derived from the centred and rotated (skewed) finite difference approximations has been shown to improve the convergence rates of these methods. In this paper, we present some theoretical convergence analysis on this preconditioner specifically applied to the linear systems resulted from these group iterative schemes in solving an elliptic boundary value problem. We will theoretically show the relationship between the spectral radiuses of the iteration matrices of the preconditioned methods which affects the rate of convergence of these methods. We will also show that the spectral radius of the preconditioned matrices is smaller than that of their unpreconditioned counterparts if the relaxation parameter is in a certain optimum range. Numerical experiments will also be presented to confirm the agreement between the theoretical and the experimental results.