Investigations on the accuracy and condition number for the method of fundamental solutions

In the applications of the method of fundamental solutions, locations of sources are treated either as variables or a priori known constants. In which, the former results in a nonlinear optimization problem and the other has to face the problem of locating sources. Theoretically, farther sources results in worse conditioning and better accuracy. In this paper, a practical procedure is provided to locate the sources for various time-independent operators, including Laplacian, Helmholtz operator, modified Helmholtz operator, and biharmonic operator. Wherein, the procedure is developed through systematic numerical experiments for relations among the accuracy, condition number, and source positions in different shapes of computational domains. In these numerical experiments, it is found that in general very good accuracy is achieved when the condition number approaches the limit of equation solver, which is a number dependent on the solution scheme and the precision. The proposed procedure is verified for both Dirichlet and Neumann boundary conditions. The general characteristics in these numerical experiments demonstrate the capability of the proposed procedure for locating sources of the method of fundamental solutions for problems without exact solutions.