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Volume 30, Issue 1
A High-Accurate Fast Poisson Solver Based on Harmonic Surface Mapping Algorithm

Jiuyang Liang, Pei Liu & Zhenli Xu

Commun. Comput. Phys., 30 (2021), pp. 210-226.

Published online: 2021-04

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  • Abstract

Poisson's equations in a cuboid are frequently solved in many scientific and engineering applications such as electric structure calculations, molecular dynamics simulations and computational astrophysics. In this paper, a fast and highly accurate algorithm is presented for the solution of the Poisson's equation in a cuboidal domain with boundary conditions of mixed type. This so-called harmonic surface mapping algorithm is a meshless algorithm which can achieve a desired order of accuracy by evaluating a body convolution of the source and the free-space Green's function within a sphere containing the cuboid, and another surface integration over the spherical surface. Numerical quadratures are introduced to approximate the integrals, resulting in the solution represented by a summation of point sources in free space, which can be accelerated by means of the fast multipole algorithm. The complexity of the algorithm is linear to the number of quadrature points, and the convergence rate can be arbitrarily high even when the source term is a piecewise continuous function.

  • AMS Subject Headings

35J08, 35Q70, 33F05, 78M16

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COPYRIGHT: © Global Science Press

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@Article{CiCP-30-210, author = {Liang , JiuyangLiu , Pei and Xu , Zhenli}, title = {A High-Accurate Fast Poisson Solver Based on Harmonic Surface Mapping Algorithm}, journal = {Communications in Computational Physics}, year = {2021}, volume = {30}, number = {1}, pages = {210--226}, abstract = {

Poisson's equations in a cuboid are frequently solved in many scientific and engineering applications such as electric structure calculations, molecular dynamics simulations and computational astrophysics. In this paper, a fast and highly accurate algorithm is presented for the solution of the Poisson's equation in a cuboidal domain with boundary conditions of mixed type. This so-called harmonic surface mapping algorithm is a meshless algorithm which can achieve a desired order of accuracy by evaluating a body convolution of the source and the free-space Green's function within a sphere containing the cuboid, and another surface integration over the spherical surface. Numerical quadratures are introduced to approximate the integrals, resulting in the solution represented by a summation of point sources in free space, which can be accelerated by means of the fast multipole algorithm. The complexity of the algorithm is linear to the number of quadrature points, and the convergence rate can be arbitrarily high even when the source term is a piecewise continuous function.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2020-0052}, url = {http://global-sci.org/intro/article_detail/cicp/18879.html} }
TY - JOUR T1 - A High-Accurate Fast Poisson Solver Based on Harmonic Surface Mapping Algorithm AU - Liang , Jiuyang AU - Liu , Pei AU - Xu , Zhenli JO - Communications in Computational Physics VL - 1 SP - 210 EP - 226 PY - 2021 DA - 2021/04 SN - 30 DO - http://doi.org/10.4208/cicp.OA-2020-0052 UR - https://global-sci.org/intro/article_detail/cicp/18879.html KW - Fast algorithm, Poisson's equation, boundary integral method, image charge, mixed boundary condition, fast multipole method. AB -

Poisson's equations in a cuboid are frequently solved in many scientific and engineering applications such as electric structure calculations, molecular dynamics simulations and computational astrophysics. In this paper, a fast and highly accurate algorithm is presented for the solution of the Poisson's equation in a cuboidal domain with boundary conditions of mixed type. This so-called harmonic surface mapping algorithm is a meshless algorithm which can achieve a desired order of accuracy by evaluating a body convolution of the source and the free-space Green's function within a sphere containing the cuboid, and another surface integration over the spherical surface. Numerical quadratures are introduced to approximate the integrals, resulting in the solution represented by a summation of point sources in free space, which can be accelerated by means of the fast multipole algorithm. The complexity of the algorithm is linear to the number of quadrature points, and the convergence rate can be arbitrarily high even when the source term is a piecewise continuous function.

Jiuyang Liang, Pei Liu & Zhenli Xu. (2021). A High-Accurate Fast Poisson Solver Based on Harmonic Surface Mapping Algorithm. Communications in Computational Physics. 30 (1). 210-226. doi:10.4208/cicp.OA-2020-0052
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