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

Part of: Numerical linear algebra

Published online by Cambridge University Press:  31 March 2011

JIANG LE ,
HUANG JIN ,
XIAO-GUANG LV  and
QING-SONG CHENG
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.

Keywords

Helmholtz equationRobbins boundary conditionsGMRES methodsine transformpreconditioner

MSC classification

Secondary: 65F10: Iterative methods for linear systems
Type
Research Article
Information
The ANZIAM Journal , Volume 52 , Issue 1 , July 2010 , pp. 87 - 100
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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