A fast singular boundary method for 3D Helmholtz equation.

This paper presents a fast singular boundary method (SBM) for three-dimensional (3D) Helmholtz equation. The SBM is a boundary-type meshless method which incorporates the advantages of the boundary element method (BEM) and the method of fundamental solutions (MFS). It is easy-to-program, and attractive to the problems with complex geometries. However, the SBM is usually limited to small-scale problems, because of the operation count of O(N3) with direct solvers or O(N2) with iterative solvers, as well as the memory requirement of O(N2). To overcome this drawback, this study makes the first attempt to employ the precorrected-FFT (PFFT) to accelerate the SBM matrix–vector multiplication at each iteration step of the GMRES for 3D Helmholtz equation. Consequently, the computational complexity can be reduced from O(N2) to O(NlogN) or O(N). Three numerical examples are successfully tested on a desktop computer. The results clearly demonstrate the accuracy and efficiency of the developed fast PFFT-SBM strategy.