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A Novel Efficient Numerical Solution of Poisson's Equation for Arbitrary Shapes in Two Dimensions

Part of:Numerical approximation and computational geometryPolytopes and polyhedra

Published online by Cambridge University Press:  02 November 2016

Zu-Hui Ma,
Weng Cho Chew and
Li Jun Jiang
Zu-Hui Ma*
Affiliation:
Department of Electrical and Electronic Engineering, The University of Hong Kong, Hong Kong
Weng Cho Chew*
Affiliation:
Department of Electrical and Computer Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 61801 USA
Li Jun Jiang*
Affiliation:
Department of Electrical and Electronic Engineering, The University of Hong Kong, Hong Kong
*
*Corresponding author. Email addresses:zuhuima@gmail.com (Z.-H. Ma), w-chew@uiuc.edu (W. C. Chew), jianglj@hku.hk (L. J. Jiang)
*Corresponding author. Email addresses:zuhuima@gmail.com (Z.-H. Ma), w-chew@uiuc.edu (W. C. Chew), jianglj@hku.hk (L. J. Jiang)
*Corresponding author. Email addresses:zuhuima@gmail.com (Z.-H. Ma), w-chew@uiuc.edu (W. C. Chew), jianglj@hku.hk (L. J. Jiang)

Abstract

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Even though there are various fast methods and preconditioning techniques available for the simulation of Poisson problems, little work has been done for solving Poisson's equation by using the Helmholtz decomposition scheme. To bridge this issue, we propose a novel efficient algorithm to solve Poisson's equation in irregular two dimensional domains for electrostatics through a quasi-Helmholtz decomposition technique—the loop-tree basis decomposition. It can handle Dirichlet, Neumann or mixed boundary problems in which the filling media can be homogeneous or inhomogeneous. A novel point of this method is to first find the electric flux efficiently by applying the loop-tree basis functions. Subsequently, the potential is obtained by finding the inverse of the gradient operator. Furthermore, treatments for both Dirichlet and Neumann boundary conditions are addressed. Finally, the validation and efficiency are illustrated by several numerical examples. Through these simulations, it is observed that the computational complexity of our proposed method almost scales as , where N is the triangle patch number of meshes. Consequently, this new algorithm is a feasible fast Poisson solver.

Keywords

Fast Poisson solverloop-tree decompositionelectrostatics

MSC classification

Secondary:52B10: Three-dimensional polytopes65D18: Computer graphics, image analysis, and computational geometry68U05: Computer graphics; computational geometry68U07: Computer-aided design

Information

Type
Research Article
Information
Communications in Computational Physics , Volume 20 , Issue 5 , November 2016, pp. 1381 - 1404
Copyright
Copyright © Global-Science Press 2016 

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