Open Access

ARTICLE

Fast Solving the Cauchy Problems of Poisson Equation in an Arbitrary Three-Dimensional Domain

Cheinshan Liu^1,2, Fajie Wang^1,3,*, Wenzheng Qu^4,*

1 Center for Numerical Simulation Software in Engineering and Sciences, College of Mechanics and Materials, Hohai University, Nanjing 211100, China.
2 Department of Mechanical and Mechatronic Engineering, National Taiwan Ocean University, Keelung 202-24, Taiwan.
3 School of Electromechanic Engineering, Qingdao University, Qingdao 266071, China.
4 Institute of Applied Mathematics, Shandong University of Technology, Zibo 255049, China.
* Corresponding author: Fajie Wang. Email: wfj1218@126.com; Wenzhen Qu. Email:quwz@sdut.edu.cn

Computer Modeling in Engineering & Sciences 2018, 114(3), 351-380. https://doi.org/10.3970/cmes.2018.114.351

Abstract

In this paper we propose a novel two-stage method to solve the three-dimensional Poisson equation in an arbitrary bounded domain enclosed by a smooth boundary. The solution is decomposed into a particular solution and a homogeneous solution. In the first stage a multiple-scale polynomial method (MSPM) is used to approximate the forcing term and then the formula of Tsai et al. [Tsai, Cheng, and Chen (2009)] is used to obtain the corresponding closed-form solution for each polynomial term. Then in the second stage we use a multiple/scale/direction Trefftz method (MSDTM) to find the solution of Laplace equation, of which the directions are uniformly distributed on a unit circle S1, and the scales are determined a priori by the collocation points on boundary. Two examples of 3D data interpolation, and several numerical examples of direct and inverse Cauchy problems in complex domain confirm the efficiency of the MSPM and the MSDTM.

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Keywords

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