An equilibrated method of fundamental solutions to choose the best source points for the Laplace equation

For the method of fundamental solutions (MFS), a trial solution is expressed as a linear combination of fundamental solutions. However, the accuracy of MFS is heavily dependent on the distribution of source points. Two distributions of source points are frequently adopted: one on a circle with a radius R, and another along an offset D to the boundary, where R and D are problem dependent constants. In the present paper, we propose a new method to choose the best source points, by using the MFS with multiple lengths Rk for the distribution of source points, which are solved from an uncoupled system of nonlinear algebraic equations. Based on the concept of equilibrated matrix, the multiple-length Rk is fully determined by the collocated points and a parameter R or D, such that the condition number of the multiple-length MFS (MLMFS) can be reduced smaller than that of the original MFS. This new technique significantly improves the accuracy of the numerical solution in several orders than the MFS with the distribution of source points using R or D. Some numerical tests for the Laplace equation confirm that the MLMFS has a good efficiency and accuracy, and the computational cost is rather cheap.