# Numerical analysis of the MFS for certain harmonic problems

Yiorgos-Sokratis Smyrlis; Andreas Karageorghis

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

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topSmyrlis, Yiorgos-Sokratis, and Karageorghis, Andreas. "Numerical analysis of the MFS for certain harmonic problems." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 38.3 (2004): 495-517. <http://eudml.org/doc/245257>.

@article{Smyrlis2004,

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 - Modélisation Mathématique et Analyse Numérique},

keywords = {method of fundamental solutions; boundary meshless methods; error bounds and convergence of the MFS; error bounds; convergence; messless methods},

language = {eng},

number = {3},

pages = {495-517},

publisher = {EDP-Sciences},

title = {Numerical analysis of the MFS for certain harmonic problems},

url = {http://eudml.org/doc/245257},

volume = {38},

year = {2004},

}

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 - Modélisation Mathématique et Analyse Numérique

PY - 2004

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; error bounds; convergence; messless methods

UR - http://eudml.org/doc/245257

ER -

## References

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- [12] Y.S. Smyrlis and A. Karageorghis, Some aspects of the method of fundamental solutions for certain harmonic problems. J. Sci. Comput. 16 (2001) 341–371. Zbl0995.65116
- [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.

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