The method of fundamental solutions and its combinations for exterior problems

Abstract

Most of the reports on the method of fundamental solutions (MFS) deal with bounded simply connected domains; only a few involve exterior problems. For exterior problems governed by Laplace's equation, there exist two kinds of infinity conditions, (1) | u | < C and (2) u = O ( ln ρ ) . For u = O ( ln ρ ) , the traditional fundamental solutions can be used. However, for | u | ≤ C , new fundamental solutions are explored. Numerical experiments are carried out to verify the theoretical analysis. The MFS and the method of particular solutions (MPS) are classified as Trefftz methods (TM) [30] using fundamental solutions (FS) and particular solutions (PS), respectively. The remarkable advantage of the MFS over the MPS is the uniform FS: ln r = ln | PQ ¯ | , where P and Q are the solution and the source points, respectively. Hence both algorithms and programming are simple. Moreover, a crack singularity in unbounded domains (i.e., exterior problem) is also studied. A combination of the TM using both PS and FS is also employed. The numerical results of the MPS and the combination of MFS and MPS coincide with each other. The study in this paper may greatly extend the application of the MFS from bounded simply connected domains to exterior domains.