A mesh-free approach to solving the axisymmetric Poisson's equation

In this article, we extend previous work of Karageorghis and Fairweather (Int J Numer Methods Engng 44 (199), 1653–1669) on the method of fundamental solutions (MFS) for solving Laplace's equation in axisymmetric geometry to the case of Poisson's equation. When the boundary condition and source term are axisymmetric, the problem reduces to solving the Poisson's equation in cylindrical coordinates in the two‐dimensional (r, z) region of the original three‐dimensional domain S. Hence, the original boundary value problem is reduced to a two‐dimensional one. To make use of the MFS, it is necessary to calculate a particular solution, which can be subtracted off, so that the MFS can be used to solve the resulting Laplace problem. This presents a novel problem, since the axisymmetric Poisson operator does not have constant coefficients, so previous methods based on radial basis functions cannot be used. To overcome this, the source term is approximated by a two‐dimensional polynomial in r and z as in Goldberg et al. (Numer Methods Partial Differential Eq 19 (2003), 112–133). One can then obtain polynomial particular solutions by the method of undetermined coefficients. The resulting Laplace equation is then solved by the axisymmetric MFS. Numerical results are given to show the accuracy and efficiency of the proposed numerical method. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005