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boundary meshless methods

ESAIM: Mathematical Modelling and Numerical Analysis (1)

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Numerical analysis of the MFS for certain harmonic problems
Yiorgos-Sokratis Smyrlis, Andreas Karageorghis
Journal:

ESAIM: Mathematical Modelling and Numerical Analysis / Volume 38 / Issue 3 / May 2004

Published online by Cambridge University Press:

15 June 2004, pp. 495-517

Print publication:

May 2004
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The Method of Fundamental Solutions (MFS) is a boundary-typemeshless method for the solution of certain elliptic boundaryvalue problems. In this work, we investigate the properties of thematrices that arise when the MFS is applied to theDirichlet problem for Laplace's equation in a disk. In particular,we study the behaviour of the eigenvalues of these matrices andthe cases in which they vanish. Based on this, we propose amodified efficient numerical algorithm for the solution of theproblem which is applicable even in the cases when the MFS matrixmight be singular. We prove the convergence of the method foranalytic boundary data and perform a stability analysis of the methodwith respect to the distance of the singularities from the originand the number of degrees of freedom. Finally, wetest the algorithm numerically.