(untitled) (Advances in Applied Mathematics and Mechanics, 2018)
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 proposed method 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.