In this article, a hybrid numerical method, called finite element method (FEM) + dipole moment (DM), is presented for efficient solution of multiscale electromagnetic radiation and scattering problems that involve structures with fine features, such as thin-wire antennas or objects. In this method, the FEM is hybridized with the DM approach to help ease certain computational burdens, such as mesh refinement, ill-conditioning, memory overload, and long computation times, when solving multiscale problems with adequate numerical precision. Although the FEM+DM method is referred to as a single algorithm above, it is actually comprised of several different hybridization and/or implementation approaches. Two hybridization approaches are described, each tailored to the type of problem, e.g., radiation or scattering. Furthermore, both iterative and noniterative (self-consistent) implementations of the FEM+DM method are discussed. One of the important characteristics of the proposed method is that it can easily be parallelized to accelerate the speed of computation, thanks to the use of DMs for handling fine features as well as for modeling mutual interactions. The hybrid method with different hybridization/implementation approaches has been implemented in an in-house code by using graphics processing unit programming in MATLAB. Several numerical results are generated by the proposed method and are compared to those obtained from a commercial electromagnetic solver.