A Fast Multipole Method For The Transmission Problem In Acoustic

In this paper, we study the solution of the scattering acoustic waves by a penetrable bounded obstacle which implies the acoustic waves can propagate through the surface of the obstacle.We firstly present the analytical solution of this equations by using separated variable method, and then the traditional standard boundary element method will be used to solve the acoustic Helmholtz equation. We reduce the classical two-dimensional transmission problem in acoustic scattering to a system of coupled boundary integral equations (BIEs), and consider the weak formulation of the resulting equations. Uniqueness and existence results for the weak solution of corresponding variational equations are established.After that we reduce the fast multipole boundary element method based on the standard Galerkin method. The multipole factorization of the Hankel function will be given. And the system of linear equations will be simplified to a simple one according to the Hankel function’s multipole factorization and the classification of discreted boundary elements. At last, boundary element methods (BEM) based on a two-level fast multipole Galerkin schemes are employed to compute the solution of the variational equation. Numerical results are presented to verify the efficiency and accuracy of the numerical methods.