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Fast Solution for Solving the Modified Helmholtz Equation withthe Method of Fundamental Solutions

Published online by Cambridge University Press:  27 March 2015

C. S. Chen ,
Xinrong Jiang ,
Wen Chen  and
Guangming Yao
C. S. Chen
Affiliation:
Department of Engineering Mechanics, Hohai University, Nanjing, China Department of Mathematics, University of Southern Mississippi, Hattiesburg, MS, USA
Xinrong Jiang
Affiliation:
Bank of Nanjing, Nanjing, China, 210008
Wen Chen*
Affiliation:
Department of Engineering Mechanics, Hohai University, Nanjing, China
Guangming Yao
Affiliation:
Department of Mathematics, Clarkson University, Potsdam, NY, USA
*
*Corresponding author. Email addresses: cschen.math@gmail.com (C. S. Chen), hawk.xrjiang@gmail.com (X. R. Jiang), chenwen@hhu.edu.cn (W. Chen), guangmingyao@gmail.com (G. Yao)
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Abstract

The method of fundamentalsolutions (MFS)is known as aneffective boundary meshless method. However, the formulation of the MFS results in a dense and extremely ill-conditioned matrix. In this paper we investigate the MFS for solving large-scale problems for the nonhomogeneous modified Helmholtz equation. The key idea is to exploit the exponential decay of the fundamental solution of the modified Helmholtz equation, and consider a sparse or diagonal matrix instead of the original dense matrix. Hence, the homogeneous solution can be obtained efficiently and accurately. A standard two-step solution process which consists of evaluating the particular solution and the homogeneous solution is applied. Polyharmonic spline radial basis functions are employed to evaluate the particular solution. Five numerical examples in irregular domains and a large number of boundary collocation points are presented to show the simplicity and effectiveness of our approach for solving large-scale problems.

Keywords

65N1065N30Method of fundamental solutionsmethod of particular solutionsboundary meshless methodpolyharmonic splinesradial basis functionsmodified Helmholtz equation
Type
Research Article
Information
Communications in Computational Physics , Volume 17 , Issue 3 , March 2015 , pp. 867 - 886
Copyright
Copyright © Global-Science Press 2015 

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