# 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

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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 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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- R. Mathon and R.L. Johnston, The approximate solution of elliptic boundary–value problems by fundamental solutions. SIAM J. Numer. Anal.14 (1977) 638–650. Zbl0368.65058
- 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
- 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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