Coupling of Fast Multipole Method and Microlocal Discretization for the 3-D Helmholtz Equation

Abstract

We are concerned with an integral method applied to the solution of the Helmholtz equation where the linear system is solved using an iterative method. We need to perform matrix–vector products whose time and memory requirements increase as a function of the wavenumber κ. Many methods have been developed to speed up the matrix–vector product calculation or to reduce the size of the system. Microlocal discretization methods enable one to consider new systems with reduced size. Another method, the fast multipole method, is one of the most efficient and robust methods used to speed up the calculation of matrix–vector products. In this paper, a coupling of these two recently developed methods is presented. This coupling enables one to reduce CPU time very efficiently for large wavenumbers. Satisfactory numerical tests are also presented to confirm the theoretical study within a new integral formulation. Results are obtained for a sphere with a size of 26λ using a resolution based on a mesh with an average edge length of about 2λ, where λ is the wavelength. Results are also given for an industrial test case from Dassault–Aviation, the Cetaf.