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A novel hybridization of the finite element and boundary integral methods is presented for an efficient and accurate numerical analysis of electromagnetic scattering and radiation problems. The proposed method derives an adaptive numerical absorbing boundary condition (ABC) for the finite element solution based on boundary integral equations. Unlike the standard finite element boundary integral (FE BI) approach, the proposed method is free of interior resonance and produces a purely sparse system matrix, which can be solved very efficiently. Unlike the traditional finite element absorbing boundary condition (FE ABC) approach, the proposed method uses an arbitrarily-shaped truncation boundary placed very close to the scatterer/radiator to minimize the computational domain and more importantly, produce a solution that converges to the true solution of the problem.