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.

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/

[1] Circulant Matrices, John Wiley & Sons, New York (1979). | MR 543191 | Zbl 0418.15017

,[2] Acoustic and Electromagnetic Scattering Analysis Using Discrete Sources. Academic Press, New York (2000).

, and ,[3] The method of fundamental solutions for elliptic boundary value problems. Adv. Comput. Math. 9 (1998) 69-95. | Zbl 0922.65074

and ,[4] The method of fundamental solutions for scattering and radiation problems. Eng. Anal. Bound. Elem. 27 (2003) 759-769. | Zbl 1060.76649

, and ,[5] Discrete Projection Methods for Integral Equations. Computational Mechanics Publications, Southampton (1996). | MR 1445293 | Zbl 0900.65384

and ,[6] 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 0945.65130

and ,[7] Table of Integrals, Series, and Products, Academic Press, London (1980). | Zbl 0521.33001

and ,[8] A mathematical study of the charge simulation method II. J. Fac. Sci., Univ. of Tokyo, Sect. 1A, Math. 36 (1989) 135-162. | Zbl 0681.65081

,[9] A mathematical study of the charge simulation method I. J. Fac. Sci., Univ. of Tokyo, Sect. 1A, Math. 35 (1988) 507-518. | Zbl 0662.65100

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

,[11] The approximate solution of elliptic boundary-value problems by fundamental solutions. SIAM J. Numer. Anal. 14 (1977) 638-650. | Zbl 0368.65058

and ,[12] Some aspects of the method of fundamental solutions for certain harmonic problems. J. Sci. Comput. 16 (2001) 341-371. | Zbl 0995.65116

and ,[13] Numerical analysis of the MFS for certain harmonic problems. Technical Report TR/04/2003, Dept. of Math. & Stat., University of Cyprus.

and ,