A fictitious domain method for solving acoustic scattering problems is considered. A non-symmetric complex-valued augmentation of the stiffness matrix is proposed. The augmentation block corresponds to the same wave operator inside the scatterer with the absorbing boundary conditions posed on its boundary. An iterative method in a subspace of h-harmonic functions is used. The arithmetical complexity of each step of the method (except an initialization step) depends on the number of mesh nodes on the scatterer boundary. The numerous experiments justify a good convergence rate of the method.