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This paper presents, for the first time in the engineering community, a symmetric coupling between the finite element and integral equation methods (FEM-IE) for solving three-dimensional unbounded radiation and scattering problems. The proposed FEM-IE is based on the E-field vector Helmholtz equation. Curl-conforming vector finite elements are used to discretize the interior region, whereas the divergence-conforming surface elements are utilized in the IE truncation surface. The symmetry in the IE part is restored through the application of the Calderon-projector. Moreover, the IE computations are accelerated using a single level QR algorithm. This reduces both memory and computational time. Furthermore it allows the use of different Green's functions for the exterior problem, with only minor modifications on the algorithm. The resulted system of equations is solved with a very efficient preconditioned conjugate gradient (PCCG) with a p-Multiplicative Schwarz preconditioner.