Volume 10, Issue 3
A Two-Constraint Method for Appropriate Determination of the Configuration of Source and Collocation Points in the Method of Fundamental Solutions for 2D Laplace Equation

M. R. Hematiyan, A. Haghighi & A. Khosravifard

Adv. Appl. Math. Mech., 10 (2018), pp. 554-580.

Published online: 2018-10

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  • Abstract

Proper positioning of collocation and source points is one of the major issues in the development of the method of fundamental solutions (MFS). In this paper, two constraints for appropriate determination of the location of collocation and source points in the MFS for two-dimensional problems are introduced. The first constraint is introduced to make sure that the solution of the problem has no oscillation between two adjacent collocation points on the boundary. Imposing the second constraint improves the condition of the generated system of equations. In other words, the second constraint reduces the condition number of the MFS system of equations. In this method, no optimization procedure is carried out. The proposed method is formulated for the Laplace problem; however, it can be developed for other problems as well. The accuracy and effectiveness of the proposedmethod is demonstrated by presenting several numerical examples. It is shown that boundary conditions with a sharp variation of the field variable can be well handled by the presented method. Moreover, it has been found that problems with a concave or re-entrant corner can be efficiently modelled by the proposed two-constraint method.

  • Keywords

Method of fundamental solutions, location of source points, location of collocation points, location parameter, condition number.

  • AMS Subject Headings

35A08, 65N80, 65K05, 15A12

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COPYRIGHT: © Global Science Press

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@Article{AAMM-10-554, author = {}, title = {A Two-Constraint Method for Appropriate Determination of the Configuration of Source and Collocation Points in the Method of Fundamental Solutions for 2D Laplace Equation}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2018}, volume = {10}, number = {3}, pages = {554--580}, abstract = {Proper positioning of collocation and source points is one of the major issues in the development of the method of fundamental solutions (MFS). In this paper, two constraints for appropriate determination of the location of collocation and source points in the MFS for two-dimensional problems are introduced. The first constraint is introduced to make sure that the solution of the problem has no oscillation between two adjacent collocation points on the boundary. Imposing the second constraint improves the condition of the generated system of equations. In other words, the second constraint reduces the condition number of the MFS system of equations. In this method, no optimization procedure is carried out. The proposed method is formulated for the Laplace problem; however, it can be developed for other problems as well. The accuracy and effectiveness of the proposedmethod is demonstrated by presenting several numerical examples. It is shown that boundary conditions with a sharp variation of the field variable can be well handled by the presented method. Moreover, it has been found that problems with a concave or re-entrant corner can be efficiently modelled by the proposed two-constraint method.}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.OA-2016-0065}, url = {http://global-sci.org/intro/article_detail/aamm/12225.html} }
TY - JOUR T1 - A Two-Constraint Method for Appropriate Determination of the Configuration of Source and Collocation Points in the Method of Fundamental Solutions for 2D Laplace Equation JO - Advances in Applied Mathematics and Mechanics VL - 3 SP - 554 EP - 580 PY - 2018 DA - 2018/10 SN - 10 DO - http://dor.org/10.4208/aamm.OA-2016-0065 UR - https://global-sci.org/intro/aamm/12225.html KW - Method of fundamental solutions, location of source points, location of collocation points, location parameter, condition number. AB - Proper positioning of collocation and source points is one of the major issues in the development of the method of fundamental solutions (MFS). In this paper, two constraints for appropriate determination of the location of collocation and source points in the MFS for two-dimensional problems are introduced. The first constraint is introduced to make sure that the solution of the problem has no oscillation between two adjacent collocation points on the boundary. Imposing the second constraint improves the condition of the generated system of equations. In other words, the second constraint reduces the condition number of the MFS system of equations. In this method, no optimization procedure is carried out. The proposed method is formulated for the Laplace problem; however, it can be developed for other problems as well. The accuracy and effectiveness of the proposedmethod is demonstrated by presenting several numerical examples. It is shown that boundary conditions with a sharp variation of the field variable can be well handled by the presented method. Moreover, it has been found that problems with a concave or re-entrant corner can be efficiently modelled by the proposed two-constraint method.
M. R. Hematiyan, A. Haghighi & A. Khosravifard. (2020). A Two-Constraint Method for Appropriate Determination of the Configuration of Source and Collocation Points in the Method of Fundamental Solutions for 2D Laplace Equation. Advances in Applied Mathematics and Mechanics. 10 (3). 554-580. doi:10.4208/aamm.OA-2016-0065
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