The type of mesh termination used in the context of finite element formulations plays a major role on the efficiency and accuracy of the field solution. In this work we evaluate the performance of an absorbing boundary condition and an artificial absorber (a new concept) for terminating the finite element mesh. This analysis is done in connection with the problem of scattering by a finite slot array in a thick ground plane. The two approximate mesh truncation schemes are compared with the exact finite element - boundary integral (FEM-BI) method in terms of accuracy and efficiency. It is demonstrated that both approximate truncation schemes yield reasonably accurate results even when the mesh is extended only 0.3 wavelengths away from the array aperture. However, the artificial absorber termination method leads to a substantially more efficient solution. Moreover, it is shown that the FEM-BI method remains quite competitive with the FEM-artificial absorber method when the fast Fourier transform is used for computing the matrixvector products in the iterative solution algorithm. These conclusions are indeed surprising and of major importance in electromagnetic simulations based on the finite element method.