no. 3
Numerical analysis of the MFS for certain harmonic problems
ESAIM: Modélisation mathématique et analyse numérique, Tome 38 (2004) no. 3, pp. 495-517.

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.

DOI : 10.1051/m2an:2004023
Classification : Primary 65N12, 65N38, Secondary 65N15, 65T50, 65Y99
Mots clés : method of fundamental solutions, boundary meshless methods, error bounds and convergence of the MFS 
@article{M2AN_2004__38_3_495_0,
     author = {Smyrlis, Yiorgos-Sokratis and Karageorghis, Andreas},
     title = {Numerical analysis of the {MFS} for certain harmonic problems},
     journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
     pages = {495--517},
     publisher = {EDP-Sciences},
     volume = {38},
     number = {3},
     year = {2004},
     doi = {10.1051/m2an:2004023},
     mrnumber = {2075757},
     zbl = {1079.65108},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/m2an:2004023/}
}
TY  - JOUR
AU  - Smyrlis, Yiorgos-Sokratis
AU  - Karageorghis, Andreas
TI  - Numerical analysis of the MFS for certain harmonic problems
JO  - ESAIM: Modélisation mathématique et analyse numérique
PY  - 2004
SP  - 495
EP  - 517
VL  - 38
IS  - 3
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/m2an:2004023/
DO  - 10.1051/m2an:2004023
LA  - en
ID  - M2AN_2004__38_3_495_0
ER  - 
%0 Journal Article
%A Smyrlis, Yiorgos-Sokratis
%A Karageorghis, Andreas
%T Numerical analysis of the MFS for certain harmonic problems
%J ESAIM: Modélisation mathématique et analyse numérique
%D 2004
%P 495-517
%V 38
%N 3
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/m2an:2004023/
%R 10.1051/m2an:2004023
%G en
%F M2AN_2004__38_3_495_0
Smyrlis, Yiorgos-Sokratis; Karageorghis, Andreas. Numerical analysis of the MFS for certain harmonic problems. ESAIM: Modélisation mathématique et analyse numérique, Tome 38 (2004) no. 3, pp. 495-517. doi : 10.1051/m2an:2004023. http://www.numdam.org/articles/10.1051/m2an:2004023/

[1] P.J. Davis, Circulant Matrices, John Wiley & Sons, New York (1979). | MR | Zbl

[2] A. Doicu, Y. Eremin and T. Wriedt, Acoustic and Electromagnetic Scattering Analysis Using Discrete Sources. Academic Press, New York (2000).

[3] G. Fairweather and A. Karageorghis, The method of fundamental solutions for elliptic boundary value problems. Adv. Comput. Math. 9 (1998) 69-95. | Zbl

[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. | Zbl

[5] M.A. Golberg and C.S. Chen, Discrete Projection Methods for Integral Equations. Computational Mechanics Publications, Southampton (1996). | MR | Zbl

[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. | Zbl

[7] I.S. Gradshteyn and I.M. Ryzhik, Table of Integrals, Series, and Products, Academic Press, London (1980). | Zbl

[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. | Zbl

[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. | Zbl

[10] J.A. Kolodziej, Applications of the Boundary Collocation Method in Applied Mechanics, Wydawnictwo Politechniki Poznanskiej, Poznan (2001) (In Polish).

[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. | Zbl

[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. | Zbl

[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.

Cité par Sources :