Volume 24, Issue 4
Fast Finite Element Method for the Three-Dimensional Poisson Equation in Infinite Domains

Xiang Ma & Chunxiong Zheng

Commun. Comput. Phys., 24 (2018), pp. 1101-1120.

Published online: 2018-06

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  • Abstract

We aim at a fast finite element method for the Poisson equation in threedimensional infinite domains. Both the exterior and strip-tail problems are considered. By introducing a suitable artificial boundary and imposing the exact boundary condition of Dirichlet-to-Neumann (DtN) type, we reduce the original infinite domain problem into a truncated finite domain problem. The point is how to efficiently implement this exact artificial boundary condition. The traditional modal expansion method is hard to apply for the strip-tail problem with a general cross section. We develop a fast algorithm based on the Padé approximation for the square root function involved in the exact artificial boundary condition. The most remarkable advantage of our method is that it is unnecessary to compute the full eigen system associated with the LaplaceBeltrami operator on the artificial boundary. Besides, compared with the modal expansion method, the computational cost of the DtN mapping is significantly reduced. We perform a complete numerical analysis on the fast algorithm. Some numerical examples are presented to demonstrate the effectiveness of the proposed method.

  • Keywords

Infinite domain problems, exact artificial boundary conditions, fast algorithms.

  • AMS Subject Headings

35A35, 65N12, 65N30

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CiCP-24-1101, author = {}, title = {Fast Finite Element Method for the Three-Dimensional Poisson Equation in Infinite Domains}, journal = {Communications in Computational Physics}, year = {2018}, volume = {24}, number = {4}, pages = {1101--1120}, abstract = {

We aim at a fast finite element method for the Poisson equation in threedimensional infinite domains. Both the exterior and strip-tail problems are considered. By introducing a suitable artificial boundary and imposing the exact boundary condition of Dirichlet-to-Neumann (DtN) type, we reduce the original infinite domain problem into a truncated finite domain problem. The point is how to efficiently implement this exact artificial boundary condition. The traditional modal expansion method is hard to apply for the strip-tail problem with a general cross section. We develop a fast algorithm based on the Padé approximation for the square root function involved in the exact artificial boundary condition. The most remarkable advantage of our method is that it is unnecessary to compute the full eigen system associated with the LaplaceBeltrami operator on the artificial boundary. Besides, compared with the modal expansion method, the computational cost of the DtN mapping is significantly reduced. We perform a complete numerical analysis on the fast algorithm. Some numerical examples are presented to demonstrate the effectiveness of the proposed method.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.2018.hh80.04}, url = {http://global-sci.org/intro/article_detail/cicp/12320.html} }
TY - JOUR T1 - Fast Finite Element Method for the Three-Dimensional Poisson Equation in Infinite Domains JO - Communications in Computational Physics VL - 4 SP - 1101 EP - 1120 PY - 2018 DA - 2018/06 SN - 24 DO - http://dor.org/10.4208/cicp.2018.hh80.04 UR - https://global-sci.org/intro/cicp/12320.html KW - Infinite domain problems, exact artificial boundary conditions, fast algorithms. AB -

We aim at a fast finite element method for the Poisson equation in threedimensional infinite domains. Both the exterior and strip-tail problems are considered. By introducing a suitable artificial boundary and imposing the exact boundary condition of Dirichlet-to-Neumann (DtN) type, we reduce the original infinite domain problem into a truncated finite domain problem. The point is how to efficiently implement this exact artificial boundary condition. The traditional modal expansion method is hard to apply for the strip-tail problem with a general cross section. We develop a fast algorithm based on the Padé approximation for the square root function involved in the exact artificial boundary condition. The most remarkable advantage of our method is that it is unnecessary to compute the full eigen system associated with the LaplaceBeltrami operator on the artificial boundary. Besides, compared with the modal expansion method, the computational cost of the DtN mapping is significantly reduced. We perform a complete numerical analysis on the fast algorithm. Some numerical examples are presented to demonstrate the effectiveness of the proposed method.

Xiang Ma & Chunxiong Zheng. (2020). Fast Finite Element Method for the Three-Dimensional Poisson Equation in Infinite Domains. Communications in Computational Physics. 24 (4). 1101-1120. doi:10.4208/cicp.2018.hh80.04
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