Volume 20, Issue 5
A Novel Efficient Numerical Solution of Poisson's Equation for Arbitrary Shapes in Two Dimensions

Zu-Hui Ma, Weng Cho Chew & Li Jun Jiang

Commun. Comput. Phys., 20 (2016), pp. 1381-1404.

Published online: 2018-04

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  • Abstract

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 $\mathcal{O}$$(N)$, where $N$ is the triangle patch number of meshes. Consequently, this new algorithm is a feasible fast Poisson solver.

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@Article{CiCP-20-1381, author = {}, title = {A Novel Efficient Numerical Solution of Poisson's Equation for Arbitrary Shapes in Two Dimensions}, journal = {Communications in Computational Physics}, year = {2018}, volume = {20}, number = {5}, pages = {1381--1404}, abstract = {

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 $\mathcal{O}$$(N)$, where $N$ is the triangle patch number of meshes. Consequently, this new algorithm is a feasible fast Poisson solver.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.230813.291113a}, url = {http://global-sci.org/intro/article_detail/cicp/11194.html} }
TY - JOUR T1 - A Novel Efficient Numerical Solution of Poisson's Equation for Arbitrary Shapes in Two Dimensions JO - Communications in Computational Physics VL - 5 SP - 1381 EP - 1404 PY - 2018 DA - 2018/04 SN - 20 DO - http://doi.org/10.4208/cicp.230813.291113a UR - https://global-sci.org/intro/article_detail/cicp/11194.html KW - AB -

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 $\mathcal{O}$$(N)$, where $N$ is the triangle patch number of meshes. Consequently, this new algorithm is a feasible fast Poisson solver.

Zu-Hui Ma, Weng Cho Chew & Li Jun Jiang. (2020). A Novel Efficient Numerical Solution of Poisson's Equation for Arbitrary Shapes in Two Dimensions. Communications in Computational Physics. 20 (5). 1381-1404. doi:10.4208/cicp.230813.291113a
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