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A novel technique is introduced for an efficient and rigorous solution of electromagnetic scattering problems from faceted bodies. This method is based on the use of analytically derived characteristic basis functions (CBFs), whose use preserves some of the desired features of the asymptotic methods. The CBFs are used to construct a matrix equation by imposing the boundary conditions on the scatterer in a numerically rigorous way via the Galerkin method, a feature unavailable in asymptotic methods. Electrically large problems can be handled by using the CBF approach in a computationally efficient manner, both in terms of time and memory. The proposed method is shown to yield good results for two-dimensional faceted bodies. In addition, it can be extended to scattering problems involving three-dimensional faceted bodies.