We propose an inverse scheme for acoustic source localization based on solving the corresponding partial differential equation in the frequency domain (Helmholtz equation) by applying the Finite Element (FE) method. This allows us to fully take into account the actual boundary conditions as given in the measurement setup. To recover the source locations, an inverse scheme based on a sparsity promoting Tikhonov functional to match measured (microphone signals) and simulated pressure is proposed. Since the differential operators for the state equation and the adjoint equation are the same, the FE system matrix for both partial differential equations is the same, which results in a computational highly efficient solution process. Furthermore, the computational time does not depend on the number of microphones nor on the assumed number of possible sources. Finally, the inverse scheme results in a source map both for amplitude and phase and in addition the reconstructed acoustic field is provided. The properties of this inverse scheme and its applicability to source localization in the low frequency range will be demonstrated.