Numerical analysis of the MFS for certain harmonic problems
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 38 (2004) no. 3, p. 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 : https://doi.org/10.1051/m2an:2004023
Classification:  Primary 65N12,  65N38,  Secondary 65N15,  65T50,  65Y99
Keywords: 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: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {38},
     number = {3},
     year = {2004},
     pages = {495-517},
     doi = {10.1051/m2an:2004023},
     zbl = {1079.65108},
     mrnumber = {2075757},
     language = {en},
     url = {http://www.numdam.org/item/M2AN_2004__38_3_495_0}
}
Smyrlis, Yiorgos-Sokratis; Karageorghis, Andreas. Numerical analysis of the MFS for certain harmonic problems. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 38 (2004) no. 3, pp. 495-517. doi : 10.1051/m2an:2004023. http://www.numdam.org/item/M2AN_2004__38_3_495_0/

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