Numerical solution of three-dimensional Laplacian problems using the multiple scale Trefftz method Cheng-Yu Ku a,n , Chung-Lun Kuo b , Chia-Ming Fan a , Chein-Shan Liu c , Pai-Chen Guan b a Department of Harbor and River Engineering, National Taiwan Ocean University, Keelung, Taiwan b Department of Systems Engineering and Naval Architecture, National Taiwan Ocean University, Keelung, Taiwan c Department of Civil Engineering, National Taiwan University, Taipei, Taiwan article info Article history: Received 13 March 2014 Received in revised form 17 July 2014 Accepted 18 August 2014 Keywords: Trefftz method Ill-conditioned The dynamical Jacobian-inverse free method The multiple scale Three-dimensional abstract This paper proposes the numerical solution of three-dimensional Laplacian problems based on the multiple scale Trefftz method with the incorporation of the dynamical Jacobian-inverse free method. A numerical solution for three-dimensional Laplacian problems was approximated by superpositioning T-complete functions formulated from 18 independent functions satisfying the governing equation in the cylindrical coordinate system. To mitigate a severely ill-conditioned system of linear equations, this study adopted the newly developed multiple scale Trefftz method and the dynamical Jacobian-inverse free method. Numerical solutions were conducted for problems involving three-dimensional ground- water flow problems enclosed by a cuboid-type domain, a peanut-type domain, a sphere domain, and a cylindrical domain. The results revealed that the proposed method can obtain accurate numerical solutions for three-dimensional Laplacian problems, yielding a superior convergence in numerical stability to that of the conventional Trefftz method. & 2014 Elsevier Ltd. All rights reserved. 1. Introduction Proposed by Trefftz in 1926 [1], the Trefftz method is a meshless numerical method for solving boundary value problems where approximate solutions are expressed as a linear combina- tion of functions automatically satisfy governing equations. According to Kita and Kamiya [2], Trefftz methods are classified as either direct or indirect formulations. Unknown coefficients are determined by matching boundary conditions. Li et al. [3] pro- vided a comprehensive comparison of the Trefftz method, colloca- tion, and other boundary methods, concluding that the collocation Trefftz method (CTM) is the simplest algorithm and provides the most accurate solutions with optimal numerical stability. Applica- tions of the Trefftz method in engineering problems, such as Laplace and biharmonic equations [4] and the two-dimensional boundary detection problem [5], have been reported. The Trefftz method has recently become popular, since it is a numerical method for easily and rapidly solving the boundary value problems. Kita et al. [6] describe the application of the Trefftz method to the solution of three-dimensional Poisson equation. An inhomogeneous term containing the unknown func- tion is approximated with a polynomial function in the Cartesian coordinates to determine the particular solution for the Poisson equation. Due to the complexity, most of the applications of the Trefftz method are still based on two-dimensional problems [7]. In the literature, to the best knowledge of the authors of this article, the formulation of Trefftz method based on the cylindrical coordi- nate system has not been found yet. In this study, we present the numerical solution of three-dimensional Laplacian problems by the collocation Trefftz method based on the cylindrical coordinate system. In the present formulation, the unknown solution is approxi- mated by superpositioning the T-complete functions satisfying the governing equation in the cylindrical coordinate system. The T-complete functions are composed of a set of linearly indepen- dent vectors. The basis for the T-complete functions includes 18 linearly independent functions. For the indirect Trefftz formula- tion, the solution is expressed as the linear combination of these basis functions. Because using conventional CTMs results in extremely ill-conditioned linear equation systems, particularly when solving three-dimensional Laplacian problems, the resulting numerical solutions may be unstable. In order to obtain an accurate solution of the linear equations, special techniques [8], e.g., the Tikhonov regularization, the singular value decomposition, conditioning by a suitable preconditioner, and truncated singular value decomposition, may be required. Li et al. [9–11] have studied on the effective condition number for collocation methods and the stability analysis was made for the Contents lists available at ScienceDirect journal homepage: www.elsevier.com/locate/enganabound Engineering Analysis with Boundary Elements http://dx.doi.org/10.1016/j.enganabound.2014.08.007 0955-7997/& 2014 Elsevier Ltd. All rights reserved. n Corresponding author. E-mail address: chkst26@mail.ntou.edu.tw (C.-Y. Ku). Engineering Analysis with Boundary Elements 50 (2015) 157–168