A fast singular boundary method for 3D Helmholtz equation. (15th January 2019)

• Record Type: Journal Article
• Authors: Li, Weiwei
• Is Part Of: Computers & mathematics with applications. Volume 77:issue 2(2019)
• Page Start: 525
• Page End: 535
• Publication Date: 2019-01-15
• Subjects: Singular boundary method -- Precorrected-FFT -- Fast algorithm -- Three-dimensional Helmholtz problems; Electronic data processing -- Periodicals; Mathematics -- Data processing -- Periodicals; 510.28541
• DOI: 10.1016/j.camwa.2018.09.055
• Languages: English
• ISSNs: 0898-1221

Abstract

Abstract: 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 ( N 3 ) with direct solvers or O ( N 2 ) with iterative solvers, as well as the memory requirement of O ( N 2 ) . 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 ( N 2 ) to O ( N l o g N ) 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.