Numerical analysis of the MFS for certain harmonic problems

Yiorgos-Sokratis Smyrlis; Andreas Karageorghis

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

  • Volume: 38, Issue: 3, page 495-517
  • ISSN: 0764-583X

Abstract

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The Method of Fundamental Solutions (MFS) is a boundary-type meshless method for the solution of certain elliptic boundary value problems. In this work, we investigate the properties of the matrices that arise when the MFS is applied to the Dirichlet problem for Laplace's equation in a disk. In particular, we study the behaviour of the eigenvalues of these matrices and the cases in which they vanish. Based on this, we propose a modified efficient numerical algorithm for the solution of the problem which is applicable even in the cases when the MFS matrix might be singular. We prove the convergence of the method for analytic boundary data and perform a stability analysis of the method with respect to the distance of the singularities from the origin and the number of degrees of freedom. Finally, we test the algorithm numerically.

How to cite

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Smyrlis, Yiorgos-Sokratis, and Karageorghis, Andreas. "Numerical analysis of the MFS for certain harmonic problems." ESAIM: Mathematical Modelling and Numerical Analysis 38.3 (2010): 495-517. <http://eudml.org/doc/194225>.

@article{Smyrlis2010,
abstract = { The Method of Fundamental Solutions (MFS) is a boundary-type meshless method for the solution of certain elliptic boundary value problems. In this work, we investigate the properties of the matrices that arise when the MFS is applied to the Dirichlet problem for Laplace's equation in a disk. In particular, we study the behaviour of the eigenvalues of these matrices and the cases in which they vanish. Based on this, we propose a modified efficient numerical algorithm for the solution of the problem which is applicable even in the cases when the MFS matrix might be singular. We prove the convergence of the method for analytic boundary data and perform a stability analysis of the method with respect to the distance of the singularities from the origin and the number of degrees of freedom. Finally, we test the algorithm numerically. },
author = {Smyrlis, Yiorgos-Sokratis, Karageorghis, Andreas},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Method of fundamental solutions; boundary meshless methods; error bounds and convergence of the MFS.; method of fundamental solutions; error bounds; convergence; messless methods},
language = {eng},
month = {3},
number = {3},
pages = {495-517},
publisher = {EDP Sciences},
title = {Numerical analysis of the MFS for certain harmonic problems},
url = {http://eudml.org/doc/194225},
volume = {38},
year = {2010},
}

TY - JOUR
AU - Smyrlis, Yiorgos-Sokratis
AU - Karageorghis, Andreas
TI - Numerical analysis of the MFS for certain harmonic problems
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2010/3//
PB - EDP Sciences
VL - 38
IS - 3
SP - 495
EP - 517
AB - The Method of Fundamental Solutions (MFS) is a boundary-type meshless method for the solution of certain elliptic boundary value problems. In this work, we investigate the properties of the matrices that arise when the MFS is applied to the Dirichlet problem for Laplace's equation in a disk. In particular, we study the behaviour of the eigenvalues of these matrices and the cases in which they vanish. Based on this, we propose a modified efficient numerical algorithm for the solution of the problem which is applicable even in the cases when the MFS matrix might be singular. We prove the convergence of the method for analytic boundary data and perform a stability analysis of the method with respect to the distance of the singularities from the origin and the number of degrees of freedom. Finally, we test the algorithm numerically.
LA - eng
KW - Method of fundamental solutions; boundary meshless methods; error bounds and convergence of the MFS.; method of fundamental solutions; error bounds; convergence; messless methods
UR - http://eudml.org/doc/194225
ER -

References

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  13. Y.S. Smyrlis and A. Karageorghis, Numerical analysis of the MFS for certain harmonic problems. Technical Report TR/04/2003, Dept. of Math. & Stat., University of Cyprus.  Zbl1079.65108

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