The Linearized Euler Equations are used widely in many aerodynamic noise prediction schemes. The Linerized Euler Equations also support instability waves, which can obscure the acoustic solution. An approach to suppress the instability waves is to solve the Linearized Euler Equations in the frequency domain. Also, the frequency domain approach is far less expensive compared to the time domain approach, in terms of the computational time: though only a single frequency can be computed at one time. This paper presents the application and comparison of two unstructured grid methods for the solution of aeroacoustic problems in the frequency domain. Algorithms are developed using the Discontinuous Galerkin and the Streamline Upwind Petrov Galerkin finite element methods in two spatial dimensions. Unlike other existing frequency domain solvers, no assumption is made regarding the nature of the mean flow. The application of the two methods to a two-dimensional benchmark problem involving the suppression of instability waves is presented. The two methods are compared in terms of the computational cost, and it is concluded that the Discontinuous Galerkin method is prohibitively more expensive compared to the Streamline Upwind Petrov Galerkin method for practical aeroacoustic applications in the frequency domain.