Dirichlet-to-Neumann boundary conditions for multiple scattering in waveguides

In this paper, we study a multiple Dirichlet-to-Neumann (MDtN) boundary condition for solving a time-harmonic multiple scattering problem governed by the Helmholtz equation in waveguides that include multiple obstacles, cavities or inhomogeneities with straight waveguides placed between them. The MDtN condition is derived by analyzing analytic solutions represented by Fourier series in the straight waveguides between obstacles, cavities or inhomogeneities. The proposed method is then to remove the straight waveguides between scatterers and impose the MDtN condition on artificial boundaries resulting from domain truncation. This numerical technique can allow a great reduction of computational efforts. The well-posedness of the reduced problem with the full MDtN condition and the reduced problem with truncated MDtN conditions are established. Also the exponential convergence of approximate solutions satisfying truncated MDtN conditions will be proved.