## Abstract and Applied Analysis

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

#### 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.

#### Article information

Source
Abstr. Appl. Anal., Volume 2012 (2012), Article ID 867598, 14 pages.

Dates
First available in Project Euclid: 14 December 2012

https://projecteuclid.org/euclid.aaa/1355495860

Digital Object Identifier
doi:10.1155/2012/867598

Mathematical Reviews number (MathSciNet)
MR2969999

Zentralblatt MATH identifier
1253.65039

#### Citation

Mohd Ali, Norhashidah Hj.; Mohammed Saeed, Abdulkafi. Convergence Analysis of the Preconditioned Group Splitting Methods in Boundary Value Problems. Abstr. Appl. Anal. 2012 (2012), Article ID 867598, 14 pages. doi:10.1155/2012/867598. https://projecteuclid.org/euclid.aaa/1355495860

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