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

Abstract

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

Implications of Geometry and the Theorem of Gauss on Newtonian Gravitational Systems and a Caveat Regarding Poisson’s Equation

“…The condition of isolation is not needed if the exterior mass is distributed spheroidally and rotating. The complexity of (17) is consistent with the axisymmetric version of Poisson's equation being difficult to solve [63] even under constant density.…”

The method of particular solutions for solving axisymmetric polyharmonic and poly-Helmholtz equations

“…A comprehensive study of the analytical particular solutions of Chebyshev polynomials for polyharmonic and poly-Helmholtz operators was given by Tsai [22]. For axisymmetric operators, Chen et al [23] and Muleshkov et al [24] found the analytical particular solutions of Chebyshev polynomials for the Laplacian and Helmholtz operators, respectively. In this paper, we extend their work to axisymmetric polyharmonic and poly-Helmholtz operators using Chebyshev polynomials.…”

“…We refer readers for more details to the review papers by Fairweather and Karageorghis [1] and Golberg and Chen [2]. The MFS has also been applied for the solution of the axisymmetric Laplace equation [23,30] and the axisymmetric Helmholtz equation [31]. To obtain the axisymmetric fundamental solutions, we can integrate the non-axisymmetric fundamental solutions analytically over a ring.…”

Approximation of multivariate functions and evaluation of particular solutions using Chebyshev polynomial and trigonometric basis functions

“…the convergent rate of this approach is slow. Golberg et al [13] and Chen et al [12] approximate the source term f by applying Chebyshev interpolation so that the approximate particular solution can be evaluated. Despite the property of spectral convergence of Chebyshev interpolation, term by term monomial expansion is required so that the closed form particular solution for some particular differential operators can be derived.…”

“…(ii) The closed form approximate particular solution is difficult to derive for general differential operator L. As a result, polynomial and trigonometric approximation [12][13][14][15][16] have been re-examined to mitigate some of these difficulties. Recently, Li and Chen [14] employed the scheme of hyper-interpolation [17] for solving a series of elliptic equations and time-dependent problems.…”