This paper introduces an efficient approach for solving the Electric Field Integral Equation (EFIE) with high-order accuracy by explicitly enforcing the continuity of the impressed current densities across boundaries of the surface patch discretization. The integral operator involved is discretized via a Nystr\"om-collocation approach based on Chebyshev polynomial expansion within each patch and a closed quadrature rule is utilized such that the discretization points inside one patch coincide with those inside another patch on the shared boundary of those two patches. The continuity enforcement is achieved by constructing a mapping from those coninciding points to a vector containing unique discretization points used in the GMRES iterative solver. The proposed approach is applied to the scattering of several different geometries including a sphere, a cube, a NURBS model imported from CAD software, and a dipole structure and results are compared with the Magnetic Field Integral Equation (MFIE) and the EFIE without enforcing continuity to illustrate the effectiveness of the approach.

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