Solution of Poisson’s equation by iterative DRBEM using compactly supported, positive definite radial basis function A.H.-D. Cheng a, * , D.-L. Young b , C.-C. Tsai b a Department of Civil and Environmental Engineering, University of Delaware, Newark, DE 19716, USA b Department of Civil Engineering, National Taiwan University, Taipei, Taiwan, R.O.C. Abstract In the numerical solution of three-dimensional boundary value problems, the matrix size can be so large that it is beyond a computer’s capacity to solve it. To overcome this difficulty, an iterative dual reciprocity boundary element method (DRBEM) is developed to solve Poisson’s equation without the need of assembling a matrix. The DRBEM procedure requires that the right hand side of Poisson’s equation be approximated by a radial basis function interpolation. In the iterative solution, it is found that only compactly supported, positive definite radial basis functions lead to converged results.  2000 Elsevier Science Ltd. All rights reserved. Keywords: Boundary element method; Dual reciprocity boundary element method; Radial basis function; Iterative method; Poisson’s equation 1. Introduction In the numerical solution of complex three-dimensional problems, a large number of discrete unknowns are required to accurately represent the geometry and the solution varia- tion. When the matrix representing the linear or nonlinear system of equations is assembled, its size can be so large such that it poses difficulty for the computer to store it in the random access memory and to solve it by elimination. Consequently, the matrix size becomes the limiting factor that defines the largest problem a given computer can solve. In the finite difference method (FDM), the need for assembling a solution matrix can be circumvented by using the so-called relaxation technique pioneered by Southwell [1,2]. To apply this technique, an initial trial solution is assigned to a solution grid. The discrete values at each node is corrected in an iterative manner, until convergence is achieved. No matrix or matrix solution is needed. Because of this advantage, large size fluid dynamic problems are typically solved by the finite difference method, not by the finite element method (FEM). It appears that this iterative solution idea can be extended to the boundary element method (BEM). A search in the literature finds a number of BEM solu- tions that utilize iterative techniques [3–7]. However, matrices were assembled in those implementations. Iterative techniques were used only to invert the matrices. These methods do not meet our definition. Iterative BEMs that do not assemble solution matrices do exist. To our knowledge, the first such attempt was made by Cahan, et al. [8] for solving Laplace’s equation based on the direct BEM formulation. Later, an improved version was presented by Cahan and Lafe [9]. An iterative BEM based on the indirect formulation was devised for solving the governing equations of stochastic bound- ary value problems [10,11]. In those stochastic problems, not only the mean, but also the covariances are obtained. For a boundary geometry discretized into N nodes, the number of unknown covariances is N 2 . The matrix, if assembled, would be of the size N 2 × N 2 : These unusually large sizes have necessitated the use of an iterative technique. In this paper, we shall revisit the iterative BEM with the goal of solving Poisson’s equation. The inhomogeneous right-hand side is treated by the dual reciprocity boundary element method (DRBEM) [12]. The underlying reason for the current practice is to construct an efficient algorithm to solve three-dimensional fluid dynamics problems governed by Navier–Stokes equations. By a velocity– vorticity formulation and a time-marching scheme, Navier–Stokes equations can be transformed into a number of Poisson’s and Helmholtz-type equations. The iterative DRBEM can then be implemented for these equations. In this paper, however, only two-dimensional Poisson’s equations are solved as a demonstration of the methodology. Engineering Analysis with Boundary Elements 24 (2000) 549–557 0955-7997/00/$ - see front matter  2000 Elsevier Science Ltd. All rights reserved. PII: S0955-7997(00)00035-7 www.elsevier.com/locate/enganabound * Corresponding author. Tel.: +1-302-831-2442; fax: +1-302-831-3640. E-mail address: cheng@chaos.ce.udel.edu (A.H.-D. Cheng).