Matrix Representation of the Fast Multipole Method of Scalar Boundary Elements

Abstract

The fast multipole boundary element method finds applications in computer simulation used for a wide variety of electromagnetic and acoustic phenomena. FMM is a tool for the boundary element method optimization. The paper proposes a block-matrix representation of multipole operators for the fast multipole boundary element method. A modification of the FMM is proposed using such a representation of the multipole operators. One of the advantages of the offered method is that the system of linear algebraic equations occurring after its use can be solved by direct methods. Moreover, it is possible to use non-specialized preconditioners in iterative methods to solve SLAE. The matrix assembled in the offered method can be used to precondition the SLAE, obtained by the fast multipole boundary element method, without the proposed modification. The article describes an algorithm for a simple method to obtain the matrix form of multipole operators. A comparison of the fast multipole boundary element method without offered modification and with block-diagonal preconditioning and with the preconditioning proposed in the article is given. During the comparison, harmonics of different orders were used to construct the preconditioner. The use of the proposed method as a preconditioner makes the SLAE preconditioning for the FMM without offered modification of boundary elements relatively simple and sufficiently high quality, which accelerates the convergence of iterative methods.