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A PRECONDITIONED METHOD FOR THE SOLUTION OF THE ROBBINS PROBLEM FOR THE HELMHOLTZ EQUATION

Published online by Cambridge University Press:  31 March 2011

JIANG LE*
Affiliation:
School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, Sichuan, PR China (email: jiangle@hhit.edu.cn, huangjin12345@163.com) School of Science, HuaiHai Institute of Technology, Lianyungang, Jiangsu, PR China (email: xiaoguanglv@126.com, qingsongcheng@hhit.edu.cn)
HUANG JIN
Affiliation:
School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, Sichuan, PR China (email: jiangle@hhit.edu.cn, huangjin12345@163.com)
XIAO-GUANG LV
Affiliation:
School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, Sichuan, PR China (email: jiangle@hhit.edu.cn, huangjin12345@163.com) School of Science, HuaiHai Institute of Technology, Lianyungang, Jiangsu, PR China (email: xiaoguanglv@126.com, qingsongcheng@hhit.edu.cn)
QING-SONG CHENG
Affiliation:
School of Science, HuaiHai Institute of Technology, Lianyungang, Jiangsu, PR China (email: xiaoguanglv@126.com, qingsongcheng@hhit.edu.cn)
*
For correspondence; e-mail: jiangle@hhit.edu.cn
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Abstract

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A preconditioned iterative method for the two-dimensional Helmholtz equation with Robbins boundary conditions is discussed. Using a finite-difference method to discretize the Helmholtz equation leads to a sparse system of equations which is too large to solve directly. The approach taken in this paper is to precondition this linear system with a sine transform based preconditioner and then solve it using the generalized minimum residual method (GMRES). An analytical formula for the eigenvalues of the preconditioned matrix is derived and it is shown that the eigenvalues are clustered around 1 except for some outliers. Numerical results are reported to demonstrate the effectiveness of the proposed method.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 2011

Footnotes

This research is supported by NSFC (10871034) and Research Grant Z2008038 from HuaiHai Institute of Technology.

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