Generalized method of fundamental solutions (GMFS) for boundary value problems

In order to cope with the instability of the method of fundamental solutions (MFS), which caused by source offset, source location, or a fictitious boundary, a generalized method of fundamental solutions (GMFS) is proposed. The crucial part of the GMFS is using a generalized fundamental solution approximation (GFSA), which adopts a bilinear combination of fundamental solutions to approximate, rather than the linear combination of the MFS. Then the numerical solution of the GMFS is decided by a group of offsets corresponding to an intervention-point diffusion (IPD), instead of the MFS' offset of a single source. To demonstrate the effectiveness of the proposed approach, five numerical examples are given. The results have shown that the GMFS is more accurate, stable, and has a better convergence rate than the traditional MFS.