Numerical analysis of the MFS for certain harmonic problems

Yiorgos-Sokratis Smyrlis; Andreas Karageorghis

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2004)

  • 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 - 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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  1. [1] P.J. Davis, Circulant Matrices, John Wiley & Sons, New York (1979). Zbl0418.15017MR543191
  2. [2] A. Doicu, Y. Eremin and T. Wriedt, Acoustic and Electromagnetic Scattering Analysis Using Discrete Sources. Academic Press, New York (2000). Zbl0948.78007
  3. [3] G. Fairweather and A. Karageorghis, The method of fundamental solutions for elliptic boundary value problems. Adv. Comput. Math. 9 (1998) 69–95. Zbl0922.65074
  4. [4] G. Fairweather, A. Karageorghis and P.A. Martin, The method of fundamental solutions for scattering and radiation problems. Eng. Anal. Bound. Elem. 27 (2003) 759–769. Zbl1060.76649
  5. [5] M.A. Golberg and C.S. Chen, Discrete Projection Methods for Integral Equations. Computational Mechanics Publications, Southampton (1996). Zbl0900.65384MR1445293
  6. [6] M.A. Golberg and C.S. Chen, The method of fundamental solutions for potential, Helmholtz and diffusion problems, in Boundary Integral Methods and Mathematical Aspects, M.A. Golberg Ed., WIT Press/Computational Mechanics Publications, Boston (1999) 103–176. Zbl0945.65130
  7. [7] I.S. Gradshteyn and I.M. Ryzhik, Table of Integrals, Series, and Products, Academic Press, London (1980). Zbl0521.33001
  8. [8] M. Katsurada, A mathematical study of the charge simulation method II. J. Fac. Sci., Univ. of Tokyo, Sect. 1A, Math. 36 (1989) 135–162. Zbl0681.65081
  9. [9] M. Katsurada and H. Okamoto, A mathematical study of the charge simulation method I. J. Fac. Sci., Univ. of Tokyo, Sect. 1A, Math. 35 (1988) 507–518. Zbl0662.65100
  10. [10] J.A. Kolodziej, Applications of the Boundary Collocation Method in Applied Mechanics, Wydawnictwo Politechniki Poznanskiej, Poznan (2001) (In Polish). 
  11. [11] 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
  12. [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. [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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