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Fast Solution of Three-Dimensional Modified Helmholtz Equations by the Method of Fundamental Solutions

Part of: Partial differential equations, boundary value problems

Published online by Cambridge University Press:  21 July 2016

Ji Lin ,
C. S. Chen  and
Chein-Shan Liu
Ji Lin*
Affiliation:
State Key Laboratory of Hydrology-Water Resources and Hydraulic Engineering, International Center for Simulation Software in Engineering and Sciences, College of Mechanics and Materials, Hohai University, Nanjing 211100, China
C. S. Chen*
Affiliation:
Department of Mathematics, University of Southern Mississippi, Hattiesburg, MS 39406, USA
Chein-Shan Liu*
Affiliation:
State Key Laboratory of Hydrology-Water Resources and Hydraulic Engineering, International Center for Simulation Software in Engineering and Sciences, College of Mechanics and Materials, Hohai University, Nanjing 211100, China Department of Civil Engineering, National Taiwan University, Taipei 106-17, Taiwan
*
*Corresponding author. Email addresses:linji861103@126.com (J. Lin), cschen.math@gmail.com (C. S. Chen), liucs@ntu.edu.tw (C.-S. Liu)
*Corresponding author. Email addresses:linji861103@126.com (J. Lin), cschen.math@gmail.com (C. S. Chen), liucs@ntu.edu.tw (C.-S. Liu)
*Corresponding author. Email addresses:linji861103@126.com (J. Lin), cschen.math@gmail.com (C. S. Chen), liucs@ntu.edu.tw (C.-S. Liu)
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Abstract

This paper describes an application of the recently developed sparse scheme of the method of fundamental solutions (MFS) for the simulation of three-dimensional modified Helmholtz problems. The solution to the given problems is approximated by a two-step strategy which consists of evaluating the particular solution and the homogeneous solution. The homogeneous solution is approximated by the traditional MFS. The original dense system of the MFS formulation is condensed into a sparse system based on the exponential decay of the fundamental solutions. Hence, the homogeneous solution can be efficiently obtained. The method of particular solutions with polyharmonic spline radial basis functions and the localized method of approximate particular solutions in combination with the Gaussian radial basis function are employed to approximate the particular solution. Three numerical examples including a near singular problem are presented to show the simplicity and effectiveness of this approach.

Keywords

Method of fundamental solutionsexponential decaymodified Helmholtz problemsparse schemelocalized method of approximate particular solutionsmethod of particular solutions

MSC classification

Secondary: 65N30: Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods
Type
Research Article
Information
Communications in Computational Physics , Volume 20 , Issue 2 , August 2016 , pp. 512 - 533
Copyright
Copyright © Global-Science Press 2016 

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