Generalized Multiscale Finite Element Method for scattering problem in heterogeneous media

In this paper, we consider the scattering problem in the heterogeneous domain. The mathematical model is described by the Helmholtz equation related to wave propagation with absorbing boundary conditions. For the solution of the problem using the finite element method, we construct an unstructured fine grid that resolves perforation on the grid level. Such approximations lead to a large system of equations. To reduce the size of the discrete system, we use a multiscale approximation on a coarse grid using the Generalized Multiscale Finite Element Method. We construct a multiscale space using the solution of local spectral problems on the snapshot space in each local domain. Two types of multiscale basis functions are presented and investigated. We present numerical results for the Helmholtz problem in a heterogeneous domain with heterogeneous properties on obstacles. The proposed method is studied for different wavenumbers and numbers of the multiscale basis functions.