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The residual error incurred when numerically solving integral equations for a number of electromagnetic radiation and scattering problems is calculated with the aid of an overdetermined system. This error is systematically reduced by adaptively refining the model for the surface current. Error reduction is achieved by selectively shrinking cell dimensions (h-refinement), increasing the order of the basis functions representing the current (p-refinement), or a combination of both (hp-refinement). The correlation between residual error and surface current error is investigated and found to be high. The impact of edge singularities and curvature discontinuities is discussed.