On Solving an Acoustic Wave Problem Via Frequency-Domain Approach and Tensorial Spline Galerkin Method

In this paper, we introduce a numerical method for solving the dynamical acoustic wave propagation problem with Robin boundary conditions. The method used here is divided into two stages. In the first stage, the equations are transformed, via the Fourier Transform, into an equivalent problem for the frequency variables. This allow us to avoid a discretization of the time variable in the considered system. Existence and uniqueness for the equation in frequency-domain are given. An approximation of the acoustic density in frequency-domain approach is also proposed by using a tensorial spline finite element Galerkin method. In the second stage, a Gauss–Hermite quadrature method is used for the computation of inverse Fourier transform of the frequency acoustic density to obtain the time-dependent solution of the acoustic wave problem. Error estimates in Sobolev spaces and convergence behavior of the presented methods are studied. Several numerical test examples are given to illustrate the performance of the proposed method, effectiveness and good resolution properties for smooth and discontinuous heterogeneous solutions.